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class="vector-toc-numb">2</span> <span>Computing the polynomial</span> </div> </a> <ul id="toc-Computing_the_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_properties_of_the_polynomial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_properties_of_the_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Basic properties of the polynomial</span> </div> </a> <ul id="toc-Basic_properties_of_the_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometric_significance_of_the_polynomial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometric_significance_of_the_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Geometric significance of the polynomial</span> </div> </a> <ul id="toc-Geometric_significance_of_the_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relations_to_satellite_operations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relations_to_satellite_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relations to satellite operations</span> </div> </a> <ul id="toc-Relations_to_satellite_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alexander–Conway_polynomial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alexander–Conway_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alexander–Conway polynomial</span> </div> </a> <ul id="toc-Alexander–Conway_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Floer_homology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_Floer_homology"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Relation to Floer homology</span> </div> </a> <ul id="toc-Relation_to_Floer_homology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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</div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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invariant">knot invariant</a> which assigns a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with integer coefficients to each knot type. <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">James Waddell Alexander II</a> discovered this, the first <a href="/wiki/Knot_polynomial" title="Knot polynomial">knot polynomial</a>, in 1923. In 1969, <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a> showed a version of this polynomial, now called the <b>Alexander–Conway polynomial</b>, could be computed using a <a href="/wiki/Skein_relation" title="Skein relation">skein relation</a>, although its significance was not realized until the discovery of the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>K</i> be a <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knot</a> in the <a href="/wiki/3-sphere" title="3-sphere">3-sphere</a>. Let <i>X</i> be the infinite <a href="/wiki/Covering_space" title="Covering space">cyclic cover</a> of the <a href="/wiki/Knot_complement" title="Knot complement">knot complement</a> of <i>K</i>. This covering can be obtained by cutting the knot complement along a <a href="/wiki/Seifert_surface" title="Seifert surface">Seifert surface</a> of <i>K</i> and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a <a href="/wiki/Deck_transformation" class="mw-redirect" title="Deck transformation">covering transformation</a> <i>t</i> acting on <i>X</i>. Consider the first homology (with integer coefficients) of <i>X</i>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daf37d45ddbb3999d6395aa6cd3b2275da6cbf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.775ex; height:2.843ex;" alt="{\displaystyle H_{1}(X)}"></span>. The transformation <i>t</i> acts on the homology and so we can consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daf37d45ddbb3999d6395aa6cd3b2275da6cbf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.775ex; height:2.843ex;" alt="{\displaystyle H_{1}(X)}"></span> a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over the ring of <a href="/wiki/Laurent_polynomial" title="Laurent polynomial">Laurent polynomials</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [t,t^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [t,t^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5e4346ad5a10b8340dd3eccd0b6fd469758911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [t,t^{-1}]}"></span>. This is called the <b>Alexander invariant</b> or <b>Alexander module</b>. </p><p>The module is finitely presentable; a <a href="/w/index.php?title=Presentation_matrix&action=edit&redlink=1" class="new" title="Presentation matrix (page does not exist)">presentation matrix</a> for this module is called the <b>Alexander matrix</b>. If the number of generators, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, is less than or equal to the number of relations, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> , then we consider the ideal generated by all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\times r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>×</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\times r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cafa6704e1fd8023c39b942b2158f07f9bf1fc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.938ex; height:1.676ex;" alt="{\displaystyle r\times r}"></span> minors of the matrix; this is the zeroth <a href="/wiki/Fitting_ideal" title="Fitting ideal">Fitting ideal</a> or <b>Alexander ideal</b> and does not depend on choice of presentation matrix. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7597b176f22149f53a5abe068c50e836a703348" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.238ex; height:1.843ex;" alt="{\displaystyle r>s}"></span>, set the ideal equal to 0. If the Alexander ideal is <a href="/wiki/Principal_ideal" title="Principal ideal">principal</a>, take a generator; this is called an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm t^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm t^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7239c760d1531fe42eba9afd4b89147794e939d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.866ex; height:2.343ex;" alt="{\displaystyle \pm t^{n}}"></span>, one often fixes a particular unique form. Alexander's choice of normalization is to make the polynomial have a positive <a href="/wiki/Constant_term" title="Constant term">constant term</a>. </p><p>Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1286b7cd287dbd92e4adb75a9fcb79a0b278c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(t)}"></span>. It turns out that the Alexander polynomial of a knot is the same polynomial for the mirror image knot. In other words, it cannot distinguish between a knot and its mirror image. </p> <div class="mw-heading mw-heading2"><h2 id="Computing_the_polynomial">Computing the polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=2" title="Edit section: Computing the polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.<sup id="cite_ref-FOOTNOTEAlexander1928_3-0" class="reference"><a href="#cite_note-FOOTNOTEAlexander1928-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Take an <a href="/wiki/Oriented" class="mw-redirect" title="Oriented">oriented</a> diagram of the knot with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> crossings; there are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf4207e44c00a9312fe3e8710efbefcd6eafdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+2}"></span> regions of the knot diagram. To work out the Alexander polynomial, first one must create an <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrix</a> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,n+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,n+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b70074a8c2110e15994f888d50c3426ebd59c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle (n,n+2)}"></span>. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> rows correspond to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> crossings, and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf4207e44c00a9312fe3e8710efbefcd6eafdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+2}"></span> columns to the regions. The values for the matrix entries are either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,-1,t,-t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mo>−</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,-1,t,-t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b16a4523824f13a4fc15177ff1e427995a8609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.919ex; height:2.509ex;" alt="{\displaystyle 0,1,-1,t,-t}"></span>. </p><p>Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line. </p> <dl><dd>on the left before undercrossing: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198aea65d6b8a474eebd6271c9499a3d92096bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.648ex; height:2.176ex;" alt="{\displaystyle -t}"></span></dd> <dd>on the right before undercrossing: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></dd> <dd>on the left after undercrossing: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span></dd> <dd>on the right after undercrossing: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span></dd></dl> <p>Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrix. Depending on the columns removed, the answer will differ by multiplication by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm t^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm t^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7239c760d1531fe42eba9afd4b89147794e939d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.866ex; height:2.343ex;" alt="{\displaystyle \pm t^{n}}"></span>, where the power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is not necessarily the number of crossings in the knot. To resolve this ambiguity, divide out the largest possible power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> and multiply by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span> if necessary, so that the constant term is positive. This gives the Alexander polynomial. </p><p>The Alexander polynomial can also be computed from the <a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert matrix</a>. </p><p>After the work of J. W. Alexander, <a href="/wiki/Ralph_Fox" title="Ralph Fox">Ralph Fox</a> considered a copresentation of the knot group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(S^{3}\backslash K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi class="MJX-variant" mathvariant="normal">∖</mi> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(S^{3}\backslash K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b79a30da75e3549e802b5fc7a14dca704c24e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.993ex; height:3.176ex;" alt="{\displaystyle \pi _{1}(S^{3}\backslash K)}"></span>, and introduced non-commutative differential calculus, which also permits one to compute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1286b7cd287dbd92e4adb75a9fcb79a0b278c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(t)}"></span>.<sup id="cite_ref-FOOTNOTEFox1961_4-0" class="reference"><a href="#cite_note-FOOTNOTEFox1961-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Basic_properties_of_the_polynomial">Basic properties of the polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=3" title="Edit section: Basic properties of the polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Alexander polynomial is symmetric: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t^{-1})=\Delta _{K}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t^{-1})=\Delta _{K}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d09333a5f006fb8aa94127d77bd7c6c2dad1d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.987ex; height:3.176ex;" alt="{\displaystyle \Delta _{K}(t^{-1})=\Delta _{K}(t)}"></span> for all knots K. </p> <dl><dd>From the point of view of the definition, this is an expression of the <a href="/wiki/Poincar%C3%A9_duality" title="Poincaré duality">Poincaré Duality isomorphism</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H_{1}X}}\simeq \mathrm {Hom} _{\mathbb {Z} [t,t^{-1}]}(H_{1}X,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>≃</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H_{1}X}}\simeq \mathrm {Hom} _{\mathbb {Z} [t,t^{-1}]}(H_{1}X,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf6bfa197833e3a4291c47791212bc9ba2172b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.45ex; height:4.009ex;" alt="{\displaystyle {\overline {H_{1}X}}\simeq \mathrm {Hom} _{\mathbb {Z} [t,t^{-1}]}(H_{1}X,G)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is the quotient of the field of fractions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [t,t^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [t,t^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5e4346ad5a10b8340dd3eccd0b6fd469758911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [t,t^{-1}]}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [t,t^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [t,t^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5e4346ad5a10b8340dd3eccd0b6fd469758911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [t,t^{-1}]}"></span>, considered as a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [t,t^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [t,t^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5e4346ad5a10b8340dd3eccd0b6fd469758911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [t,t^{-1}]}"></span>-module, and where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {H_{1}X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {H_{1}X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7f7f81ec94c0ba2227feba8502f12ba05db744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.117ex; height:3.343ex;" alt="{\displaystyle {\overline {H_{1}X}}}"></span> is the conjugate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [t,t^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [t,t^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5e4346ad5a10b8340dd3eccd0b6fd469758911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.89ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [t,t^{-1}]}"></span>-module to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5837812ed607baeba52f34c458b9f87311abb29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.965ex; height:2.509ex;" alt="{\displaystyle H_{1}X}"></span> ie: as an abelian group it is identical to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5837812ed607baeba52f34c458b9f87311abb29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.965ex; height:2.509ex;" alt="{\displaystyle H_{1}X}"></span> but the covering transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> acts by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82182729910430d9930cc0cbf96a3f43050b381c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.172ex; height:2.676ex;" alt="{\displaystyle t^{-1}}"></span>.</dd></dl> <p>Furthermore, the Alexander polynomial evaluates to a unit on 1: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(1)=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(1)=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f199b76c9e4aff50d1320f83d29ca172cbf80b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.67ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(1)=\pm 1}"></span>. </p> <dl><dd>From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. More generally if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a 3-manifold such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rank(H_{1}M)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rank(H_{1}M)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edd9c6b65ecf9fa66006e2b0258611f1d0c8e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.382ex; height:2.843ex;" alt="{\displaystyle rank(H_{1}M)=1}"></span> it has an Alexander polynomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{M}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{M}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8b1ffa342b8f1db8dabbaa91cd5fa2db3f5d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle \Delta _{M}(t)}"></span> defined as the order ideal of its infinite-cyclic covering space. In this case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{M}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{M}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896c94bce4aa768da6f9f7484ca25f72330ecd76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.867ex; height:2.843ex;" alt="{\displaystyle \Delta _{M}(1)}"></span> is, up to sign, equal to the order of the torsion subgroup of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68eb0c5ab7408334b7f47f4bbe6f92292e6d53c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.428ex; height:2.509ex;" alt="{\displaystyle H_{1}M}"></span>.</dd></dl> <p>Every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Geometric_significance_of_the_polynomial">Geometric significance of the polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=4" title="Edit section: Geometric significance of the polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the Alexander ideal is principal, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab57a68adc9a6401560d799cefa9cd9c01c71fe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.539ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(t)=1}"></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the commutator subgroup of the knot group is <a href="/wiki/Perfect_group" title="Perfect group">perfect</a> (i.e. equal to its own <a href="/wiki/Commutator_subgroup" title="Commutator subgroup">commutator subgroup</a>). </p><p>For a <a href="/wiki/Topologically_slice" class="mw-redirect" title="Topologically slice">topologically slice</a> knot, the Alexander polynomial satisfies the Fox–Milnor condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)=f(t)f(t^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)=f(t)f(t^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806d0e837786c649b920155f3a6868c423ec4f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.564ex; height:3.176ex;" alt="{\displaystyle \Delta _{K}(t)=f(t)f(t^{-1})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> is some other integral Laurent polynomial. </p><p>Twice the <a href="/wiki/Seifert_surface" title="Seifert surface">knot genus</a> is bounded below by the degree of the Alexander polynomial. </p><p><a href="/wiki/Michael_Freedman" title="Michael Freedman">Michael Freedman</a> proved that a knot in the 3-sphere is <a href="/wiki/Topologically_slice" class="mw-redirect" title="Topologically slice">topologically slice</a>; i.e., bounds a "locally-flat" topological disc in the 4-ball, if the Alexander polynomial of the knot is trivial.<sup id="cite_ref-FOOTNOTEFreedmanQuinn1990_7-0" class="reference"><a href="#cite_note-FOOTNOTEFreedmanQuinn1990-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Kauffman describes the first construction of the Alexander polynomial via state sums derived from physical models. A survey of these topic and other connections with physics are given in.<sup id="cite_ref-FOOTNOTEKauffman1983_8-0" class="reference"><a href="#cite_note-FOOTNOTEKauffman1983-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEKauffman2012_9-0" class="reference"><a href="#cite_note-FOOTNOTEKauffman2012-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>There are other relations with surfaces and smooth 4-dimensional topology. For example, under certain assumptions, there is a way of modifying a smooth <a href="/wiki/4-manifold" title="4-manifold">4-manifold</a> by performing a <a href="/wiki/Surgery_theory" title="Surgery theory">surgery</a> that consists of removing a neighborhood of a two-dimensional torus and replacing it with a knot complement crossed with <i>S</i><sup>1</sup>. The result is a smooth 4-manifold homeomorphic to the original, though now the <a href="/wiki/Seiberg%E2%80%93Witten_invariant" class="mw-redirect" title="Seiberg–Witten invariant">Seiberg–Witten invariant</a> has been modified by multiplication with the Alexander polynomial of the knot.<sup id="cite_ref-FOOTNOTEFintushelStern1998_10-0" class="reference"><a href="#cite_note-FOOTNOTEFintushelStern1998-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Knots with symmetries are known to have restricted Alexander polynomials.<sup id="cite_ref-FOOTNOTEKawauchi2012symmetry_section_11-0" class="reference"><a href="#cite_note-FOOTNOTEKawauchi2012symmetry_section-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility. </p><p>If the <a href="/wiki/Knot_complement" title="Knot complement">knot complement</a> fibers over the circle, then the Alexander polynomial of the knot is known to be <i>monic</i> (the coefficients of the highest and lowest order terms are equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"></span>). In fact, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to C_{K}\to S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">→</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\to C_{K}\to S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ea675c24622f9d632454449e11c7cb56d2902c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.658ex; height:3.009ex;" alt="{\displaystyle S\to C_{K}\to S^{1}}"></span> is a fiber bundle where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a058c9a09c55f41d653d480d7f1774987e8d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.355ex; height:2.509ex;" alt="{\displaystyle C_{K}}"></span> is the knot complement, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:S\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mi>S</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:S\to S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35dedc5d30452430b6f7c17909037babb646211c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.666ex; height:2.509ex;" alt="{\displaystyle g:S\to S}"></span> represent the <a href="/wiki/Monodromy" title="Monodromy">monodromy</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)={\rm {Det}}(tI-g_{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>I</mi> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)={\rm {Det}}(tI-g_{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bb27301d7180fa5cb922d13b7e80f7f67cb61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.913ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(t)={\rm {Det}}(tI-g_{*})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{*}\colon H_{1}S\to H_{1}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>:</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>S</mi> <mo stretchy="false">→</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{*}\colon H_{1}S\to H_{1}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a791033a7ecbbe84a23b3b24e543ea20f972a3ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.781ex; height:2.509ex;" alt="{\displaystyle g_{*}\colon H_{1}S\to H_{1}S}"></span> is the induced map on homology. </p> <div class="mw-heading mw-heading2"><h2 id="Relations_to_satellite_operations">Relations to satellite operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=5" title="Edit section: Relations to satellite operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a knot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is a <a href="/wiki/Satellite_knot" title="Satellite knot">satellite knot</a> with pattern knot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63399d2b53fc472cfd349fd6167c21585d0a37f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}"></span> (there exists an embedding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S^{1}\times D^{2}\to S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:S^{1}\times D^{2}\to S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232f216b5f528824ea77de4f154141360736a133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.8ex; height:3.009ex;" alt="{\displaystyle f:S^{1}\times D^{2}\to S^{3}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=f(K')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>K</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=f(K')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d811c2de3de1f840bf80b7e080a92147acc565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.031ex; height:3.009ex;" alt="{\displaystyle K=f(K')}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\times D^{2}\subset S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⊂</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}\times D^{2}\subset S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b62ea71d96be6589ef813a4d76a66c868acb829a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.069ex; height:2.676ex;" alt="{\displaystyle S^{1}\times D^{2}\subset S^{3}}"></span> is an unknotted solid torus containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63399d2b53fc472cfd349fd6167c21585d0a37f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.779ex; height:2.509ex;" alt="{\displaystyle K'}"></span>), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)=\Delta _{f(S^{1}\times \{0\})}(t^{a})\Delta _{K'}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>K</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)=\Delta _{f(S^{1}\times \{0\})}(t^{a})\Delta _{K'}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca294c289d5498f86235c7440943af8660abf31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.96ex; height:3.343ex;" alt="{\displaystyle \Delta _{K}(t)=\Delta _{f(S^{1}\times \{0\})}(t^{a})\Delta _{K'}(t)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175ebd03e0644b0967a63d648c2843a5e883257b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.621ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {Z} }"></span> is the integer that represents <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K'\subset S^{1}\times D^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mo>′</mo> </msup> <mo>⊂</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K'\subset S^{1}\times D^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/defccc3bb0060023064894dd12f1b137e9af3035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.272ex; height:2.676ex;" alt="{\displaystyle K'\subset S^{1}\times D^{2}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}(S^{1}\times D^{2})=\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(S^{1}\times D^{2})=\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a0e8bbffeb0aba4a6b74fe28b9db6304d10378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.838ex; height:3.176ex;" alt="{\displaystyle H_{1}(S^{1}\times D^{2})=\mathbb {Z} }"></span>. </p><p>Examples: For a connect-sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K_{1}\#K_{2}}(t)=\Delta _{K_{1}}(t)\Delta _{K_{2}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi mathvariant="normal">#</mi> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K_{1}\#K_{2}}(t)=\Delta _{K_{1}}(t)\Delta _{K_{2}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cae7b5e72226c7e34e31593de541717518d7cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.825ex; height:3.009ex;" alt="{\displaystyle \Delta _{K_{1}\#K_{2}}(t)=\Delta _{K_{1}}(t)\Delta _{K_{2}}(t)}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is an untwisted <a href="/wiki/Satellite_knot" title="Satellite knot">Whitehead double</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{K}(t)=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{K}(t)=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae75bddf77f296a90f2a584feadef6b4be1813d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.347ex; height:2.843ex;" alt="{\displaystyle \Delta _{K}(t)=\pm 1}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Alexander–Conway_polynomial"><span id="Alexander.E2.80.93Conway_polynomial"></span>Alexander–Conway polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=6" title="Edit section: Alexander–Conway polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alexander proved the Alexander polynomial satisfies a skein relation. <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a> later rediscovered this in a different form and showed that the skein relation together with a choice of value on the unknot was enough to determine the polynomial. Conway's version is a polynomial in <i>z</i> with integer coefficients, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af104cc09f477673d4080039a5f70ca82c23b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.833ex; height:2.843ex;" alt="{\displaystyle \nabla (z)}"></span> and called the <b>Alexander–Conway polynomial</b> (also known as <b>Conway polynomial</b> or <b>Conway–Alexander polynomial</b>). </p><p>Suppose we are given an oriented link diagram, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{+},L_{-},L_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{+},L_{-},L_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c068537cef53c832e1388400f99b3a33892af0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.892ex; height:2.509ex;" alt="{\displaystyle L_{+},L_{-},L_{0}}"></span> are link diagrams resulting from crossing and smoothing changes on a local region of a specified crossing of the diagram, as indicated in the figure. </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Skein_(HOMFLY).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/200px-Skein_%28HOMFLY%29.svg.png" decoding="async" width="200" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/300px-Skein_%28HOMFLY%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Skein_%28HOMFLY%29.svg/400px-Skein_%28HOMFLY%29.svg.png 2x" data-file-width="300" data-file-height="160" /></a><figcaption></figcaption></figure> <p>Here are Conway's skein relations: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (O)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (O)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae4f9d03570d5c8f43a26e073def0343927d039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.779ex; height:2.843ex;" alt="{\displaystyle \nabla (O)=1}"></span> (where O is any diagram of the unknot)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2690d5a59af842c7e7496aecae7dc964c6bc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.087ex; height:2.843ex;" alt="{\displaystyle \nabla (L_{+})-\nabla (L_{-})=z\nabla (L_{0})}"></span></li></ul> <p>The relationship to the standard Alexander polynomial is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc21c36f3fdc3a8d26c960544d75e7a9e5d1b264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.038ex; height:3.176ex;" alt="{\displaystyle \Delta _{L}(t^{2})=\nabla _{L}(t-t^{-1})}"></span>. Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2644d670c7dc492c321af78d6a0532a91c2eee7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.287ex; height:2.509ex;" alt="{\displaystyle \Delta _{L}}"></span> must be properly normalized (by multiplication of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm t^{n/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm t^{n/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5c2c90f02849616102ba63ea618f19dcb13ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.51ex; height:2.843ex;" alt="{\displaystyle \pm t^{n/2}}"></span>) to satisfy the skein relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (L_{+})-\Delta (L_{-})=(t^{1/2}-t^{-1/2})\Delta (L_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (L_{+})-\Delta (L_{-})=(t^{1/2}-t^{-1/2})\Delta (L_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d852da5e0270a926d918dac85af60b4f04423f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.003ex; height:3.343ex;" alt="{\displaystyle \Delta (L_{+})-\Delta (L_{-})=(t^{1/2}-t^{-1/2})\Delta (L_{0})}"></span>. Note that this relation gives a Laurent polynomial in <i>t<sup>1/2</sup></i>. </p><p>See <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a> for an example computing the Conway polynomial of the trefoil. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_Floer_homology">Relation to Floer homology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=7" title="Edit section: Relation to Floer homology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using pseudo-holomorphic curves, Ozsváth-Szabó<sup id="cite_ref-FOOTNOTEOzsváthSzabó2004_12-0" class="reference"><a href="#cite_note-FOOTNOTEOzsváthSzabó2004-12"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and Rasmussen<sup id="cite_ref-FOOTNOTERasmussen2003_13-0" class="reference"><a href="#cite_note-FOOTNOTERasmussen2003-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> associated a bigraded abelian group, called knot Floer homology, to each isotopy class of knots. The graded <a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of knot Floer homology is the Alexander polynomial. While the Alexander polynomial gives a lower bound on the genus of a knot, <sup id="cite_ref-FOOTNOTEOzsváthSzabó2004b_14-0" class="reference"><a href="#cite_note-FOOTNOTEOzsváthSzabó2004b-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> showed that knot Floer homology detects the genus. Similarly, while the Alexander polynomial gives an obstruction to a knot complement fibering over the circle, <sup id="cite_ref-FOOTNOTENi2007_15-0" class="reference"><a href="#cite_note-FOOTNOTENi2007-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> showed that knot Floer homology completely determines when a knot complement fibers over the circle. The knot Floer homology groups are part of the Heegaard Floer homology family of invariants; see <a href="/wiki/Floer_homology" title="Floer homology">Floer homology</a> for further discussion. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. <a href="/wiki/Joan_Birman" title="Joan Birman">Joan Birman</a> mentions in her paper that Mark Kidwell brought her attention to Alexander's relation in 1970.<sup id="cite_ref-FOOTNOTEBirman1993_1-0" class="reference"><a href="#cite_note-FOOTNOTEBirman1993-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Detailed exposition of this approach about higher Alexander polynomials can be found in <a href="#CITEREFCrowellFox1963">Crowell & Fox (1963)</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 20em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBirman1993-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBirman1993_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBirman1993">Birman 1993</a>.</span> </li> <li id="cite_note-FOOTNOTEAlexander1928-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAlexander1928_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAlexander1928">Alexander 1928</a>.</span> </li> <li id="cite_note-FOOTNOTEFox1961-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFox1961_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFox1961">Fox 1961</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFKawauchi2012">Kawauchi 2012</a>, Theorem 11.5.3, p. 150. Kawauchi credits this result to Kondo, H. (1979), "Knots of unknotting number 1 and their Alexander polynomials", <i>Osaka J. Math.</i> 16: 551-559, and to Sakai, T. (1977), "A remark on the Alexander polynomials of knots", <i>Math. Sem. Notes Kobe Univ.</i> 5: 451~456.</span> </li> <li id="cite_note-FOOTNOTEFreedmanQuinn1990-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFreedmanQuinn1990_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFreedmanQuinn1990">Freedman & Quinn 1990</a>.</span> </li> <li id="cite_note-FOOTNOTEKauffman1983-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKauffman1983_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKauffman1983">Kauffman 1983</a>.</span> </li> <li id="cite_note-FOOTNOTEKauffman2012-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKauffman2012_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKauffman2012">Kauffman 2012</a>.</span> </li> <li id="cite_note-FOOTNOTEFintushelStern1998-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFintushelStern1998_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFintushelStern1998">Fintushel & Stern 1998</a>.</span> </li> <li id="cite_note-FOOTNOTEKawauchi2012symmetry_section-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKawauchi2012symmetry_section_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKawauchi2012">Kawauchi 2012</a>, symmetry section.</span> </li> <li id="cite_note-FOOTNOTEOzsváthSzabó2004-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOzsváthSzabó2004_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOzsváthSzabó2004">Ozsváth & Szabó 2004</a>.</span> </li> <li id="cite_note-FOOTNOTERasmussen2003-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERasmussen2003_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRasmussen2003">Rasmussen 2003</a>.</span> </li> <li id="cite_note-FOOTNOTEOzsváthSzabó2004b-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOzsváthSzabó2004b_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOzsváthSzabó2004b">Ozsváth & Szabó 2004b</a>.</span> </li> <li id="cite_note-FOOTNOTENi2007-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENi2007_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNi2007">Ni 2007</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=10" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAdams2004" class="citation book cs1"><a href="/wiki/Colin_Adams_(mathematician)" title="Colin Adams (mathematician)">Adams, Colin C.</a> (2004) [1994]. <i>The Knot Book: An elementary introduction to the mathematical theory of knots</i>. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3678-1" title="Special:BookSources/978-0-8218-3678-1"><bdi>978-0-8218-3678-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Knot+Book%3A+An+elementary+introduction+to+the+mathematical+theory+of+knots&rft.pub=American+Mathematical+Society&rft.date=2004&rft.isbn=978-0-8218-3678-1&rft.aulast=Adams&rft.aufirst=Colin+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span> (accessible introduction utilizing a skein relation approach)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander1928" class="citation journal cs1"><a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">Alexander, J. W.</a> (1928). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1928-030-02/S0002-9947-1928-1501429-1/S0002-9947-1928-1501429-1.pdf">"Topological Invariants of Knots and Links"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>. <b>30</b> (2): <span class="nowrap">275–</span>306. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1928-1501429-1">10.1090/S0002-9947-1928-1501429-1</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989123">1989123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Topological+Invariants+of+Knots+and+Links&rft.volume=30&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E275-%3C%2Fspan%3E306&rft.date=1928&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1928-1501429-1&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1989123%23id-name%3DJSTOR&rft.aulast=Alexander&rft.aufirst=J.+W.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1928-030-02%2FS0002-9947-1928-1501429-1%2FS0002-9947-1928-1501429-1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirman1993" class="citation journal cs1">Birman, Joan (1993). "New points of view in knot theory". <i>Bull. Amer. Math. Soc</i>. N.S. <b>28</b> (2): <span class="nowrap">253–</span>287.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=New+points+of+view+in+knot+theory&rft.volume=28&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E253-%3C%2Fspan%3E287&rft.date=1993&rft.aulast=Birman&rft.aufirst=Joan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrowellFox1963" class="citation book cs1">Crowell, Richard; <a href="/wiki/Ralph_Fox" title="Ralph Fox">Fox, Ralph</a> (1963). <i>Introduction to Knot Theory</i>. Ginn and Co. after 1977 Springer Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Knot+Theory&rft.pub=Ginn+and+Co.+after+1977+Springer+Verlag&rft.date=1963&rft.aulast=Crowell&rft.aufirst=Richard&rft.au=Fox%2C+Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFintushelStern1998" class="citation journal cs1"><a href="/wiki/Ronald_Fintushel" title="Ronald Fintushel">Fintushel, Ronald</a>; <a href="/wiki/Ronald_J._Stern" title="Ronald J. Stern">Stern, Ronald J.</a> (October 1998). "Knots, links, and 4-manifolds". <i><a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a></i>. <b>134</b> (2): <span class="nowrap">363–</span>400. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/dg-ga/9612014">dg-ga/9612014</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998InMat.134..363F">1998InMat.134..363F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002220050268">10.1007/s002220050268</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1650308">1650308</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3752148">3752148</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Knots%2C+links%2C+and+4-manifolds&rft.volume=134&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E363-%3C%2Fspan%3E400&rft.date=1998-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3752148%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1998InMat.134..363F&rft_id=info%3Aarxiv%2Fdg-ga%2F9612014&rft.issn=0020-9910&rft_id=info%3Adoi%2F10.1007%2Fs002220050268&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1650308%23id-name%3DMR&rft.aulast=Fintushel&rft.aufirst=Ronald&rft.au=Stern%2C+Ronald+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFox1961" class="citation book cs1"><a href="/wiki/Ralph_Fox" title="Ralph Fox">Fox, Ralph</a> (1961). "A quick trip through knot theory". In Fort, M.K. (ed.). <i>Proceedings of the University of Georgia Topology Institute</i>. Englewood Cliffs. N. J.: Prentice-Hall. pp. <span class="nowrap">120–</span>167. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/73203715">73203715</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+quick+trip+through+knot+theory&rft.btitle=Proceedings+of+the+University+of+Georgia+Topology+Institute&rft.place=Englewood+Cliffs.+N.+J.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E120-%3C%2Fspan%3E167&rft.pub=Prentice-Hall&rft.date=1961&rft_id=info%3Aoclcnum%2F73203715&rft.aulast=Fox&rft.aufirst=Ralph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreedmanQuinn1990" class="citation book cs1"><a href="/wiki/Michael_H._Freedman" class="mw-redirect" title="Michael H. Freedman">Freedman, Michael H.</a>; <a href="/wiki/Frank_Quinn_(mathematician)" title="Frank Quinn (mathematician)">Quinn, Frank</a> (1990). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/topologyof4manif0000free"><i>Topology of 4-manifolds</i></a></span>. Princeton Mathematical Series. Vol. 39. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08577-7" title="Special:BookSources/978-0-691-08577-7"><bdi>978-0-691-08577-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology+of+4-manifolds&rft.series=Princeton+Mathematical+Series&rft.pub=Princeton+University+Press&rft.date=1990&rft.isbn=978-0-691-08577-7&rft.aulast=Freedman&rft.aufirst=Michael+H.&rft.au=Quinn%2C+Frank&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopologyof4manif0000free&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman1983" class="citation book cs1"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis</a> (2006) [1983]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WzdbcxGdXTMC"><i>Formal Knot Theory</i></a>. Courier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45052-0" title="Special:BookSources/978-0-486-45052-0"><bdi>978-0-486-45052-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formal+Knot+Theory&rft.pub=Courier&rft.date=2006&rft.isbn=978-0-486-45052-0&rft.aulast=Kauffman&rft.aufirst=Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DWzdbcxGdXTMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman2012" class="citation book cs1"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis</a> (2012). <i>Knots and Physics</i> (4th ed.). World Scientific Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4383-00-4" title="Special:BookSources/978-981-4383-00-4"><bdi>978-981-4383-00-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Physics&rft.edition=4th&rft.pub=World+Scientific+Publishing+Company&rft.date=2012&rft.isbn=978-981-4383-00-4&rft.aulast=Kauffman&rft.aufirst=Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKawauchi2012" class="citation book cs1">Kawauchi, Akio (2012) [1996]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RkEBCAAAQBAJ"><i>A Survey of Knot Theory</i></a>. Birkhäuser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-9227-8" title="Special:BookSources/978-3-0348-9227-8"><bdi>978-3-0348-9227-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Knot+Theory&rft.pub=Birkh%C3%A4user&rft.date=2012&rft.isbn=978-3-0348-9227-8&rft.aulast=Kawauchi&rft.aufirst=Akio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRkEBCAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span> (covers several different approaches, explains relations between different versions of the Alexander polynomial)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNi2007" class="citation journal cs1">Ni, Yi (2007). "Knot Floer homology detects fibred knots". <i>Inventiones Mathematicae</i>. Invent. Math. <b>170</b> (3): <span class="nowrap">577–</span>608. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0607156">math/0607156</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007InMat.170..577N">2007InMat.170..577N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00222-007-0075-9">10.1007/s00222-007-0075-9</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119159648">119159648</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Knot+Floer+homology+detects+fibred+knots&rft.volume=170&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E577-%3C%2Fspan%3E608&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0607156&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119159648%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00222-007-0075-9&rft_id=info%3Abibcode%2F2007InMat.170..577N&rft.aulast=Ni&rft.aufirst=Yi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOzsváthSzabó2004" class="citation journal cs1"><a href="/wiki/Peter_Ozsv%C3%A1th" title="Peter Ozsváth">Ozsváth, Peter</a>; <a href="/wiki/Zolt%C3%A1n_Szab%C3%B3_(mathematician)" title="Zoltán Szabó (mathematician)">Szabó, Zoltán</a> (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aim.2003.05.001">"Holomorphic disks and knot invariants"</a>. <i><a href="/wiki/Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a></i>. <b>186</b> (1): <span class="nowrap">58–</span>116. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0209056">math/0209056</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002math......9056O">2002math......9056O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.aim.2003.05.001">10.1016/j.aim.2003.05.001</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11246611">11246611</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=Holomorphic+disks+and+knot+invariants&rft.volume=186&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E58-%3C%2Fspan%3E116&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0209056&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11246611%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.aim.2003.05.001&rft_id=info%3Abibcode%2F2002math......9056O&rft.aulast=Ozsv%C3%A1th&rft.aufirst=Peter&rft.au=Szab%C3%B3%2C+Zolt%C3%A1n&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.aim.2003.05.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOzsváthSzabó2004b" class="citation journal cs1"><a href="/wiki/Peter_Ozsv%C3%A1th" title="Peter Ozsváth">Ozsváth, Peter</a>; <a href="/wiki/Zolt%C3%A1n_Szab%C3%B3_(mathematician)" title="Zoltán Szabó (mathematician)">Szabó, Zoltán</a> (2004b). "Holomorphic disks and genus bounds". <i><a href="/wiki/Geometry_and_Topology" class="mw-redirect" title="Geometry and Topology">Geometry and Topology</a></i>. <b>8</b> (2004): <span class="nowrap">311–</span>334. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0311496">math/0311496</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fgt.2004.8.311">10.2140/gt.2004.8.311</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11374897">11374897</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geometry+and+Topology&rft.atitle=Holomorphic+disks+and+genus+bounds&rft.volume=8&rft.issue=2004&rft.pages=%3Cspan+class%3D%22nowrap%22%3E311-%3C%2Fspan%3E334&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0311496&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11374897%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2140%2Fgt.2004.8.311&rft.aulast=Ozsv%C3%A1th&rft.aufirst=Peter&rft.au=Szab%C3%B3%2C+Zolt%C3%A1n&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRasmussen2003" class="citation thesis cs1">Rasmussen, Jacob (2003). <i>Floer homology and knot complements</i> (Thesis). Harvard University. p. 6378. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0306378">math/0306378</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003math......6378R">2003math......6378R</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Floer+homology+and+knot+complements&rft.inst=Harvard+University&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0306378&rft_id=info%3Abibcode%2F2003math......6378R&rft.aulast=Rasmussen&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRolfsen1990" class="citation book cs1"><a href="/wiki/Dale_Rolfsen" class="mw-redirect" title="Dale Rolfsen">Rolfsen, Dale</a> (1990). <i>Knots and Links</i> (2nd ed.). Publish or Perish. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-914098-16-4" title="Special:BookSources/978-0-914098-16-4"><bdi>978-0-914098-16-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knots+and+Links&rft.edition=2nd&rft.pub=Publish+or+Perish&rft.date=1990&rft.isbn=978-0-914098-16-4&rft.aulast=Rolfsen&rft.aufirst=Dale&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span> (explains classical approach using the Alexander invariant; knot and link table with Alexander polynomials)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alexander_polynomial&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Alexander_invariants">"Alexander invariants"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Alexander+invariants&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAlexander_invariants&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlexander+polynomial" class="Z3988"></span></li> <li>"<a rel="nofollow" class="external text" href="https://katlas.org/wiki/Main_Page">Main Page</a>" and "<a rel="nofollow" class="external text" href="https://katlas.org/wiki/The_Alexander-Conway_Polynomial">The Alexander-Conway Polynomial</a>", <i><a href="/wiki/The_Knot_Atlas" title="The Knot Atlas">The Knot Atlas</a></i>. – knot and link tables with computed Alexander and Conway polynomials</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output 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aria-labelledby="Knot_theory_(knots_and_links)479" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Knot_theory" title="Template:Knot theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Knot_theory" title="Template talk:Knot theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Knot_theory" title="Special:EditPage/Template:Knot theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Knot_theory_(knots_and_links)479" style="font-size:114%;margin:0 4em"><a href="/wiki/Knot_theory" title="Knot theory">Knot theory</a> (<a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a> and <a href="/wiki/Link_(knot_theory)" title="Link (knot theory)">links</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hyperbolic_link" title="Hyperbolic link">Hyperbolic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Figure-eight_knot_(mathematics)" title="Figure-eight knot (mathematics)">Figure-eight</a> (4<sub>1</sub>)</li> <li><a href="/wiki/Three-twist_knot" title="Three-twist knot">Three-twist</a> (5<sub>2</sub>)</li> <li><a href="/wiki/Stevedore_knot_(mathematics)" title="Stevedore knot (mathematics)">Stevedore</a> (6<sub>1</sub>)</li> <li><a href="/wiki/6%E2%82%82_knot" class="mw-redirect" title="6₂ knot">6<sub>2</sub></a></li> <li><a href="/wiki/6%E2%82%83_knot" class="mw-redirect" title="6₃ knot">6<sub>3</sub></a></li> <li><a href="/wiki/7%E2%82%84_knot" class="mw-redirect" title="7₄ knot">Endless</a> (7<sub>4</sub>)</li> <li><a href="/wiki/Carrick_mat" title="Carrick mat">Carrick mat</a> (8<sub>18</sub>)</li> <li><a href="/wiki/Perko_pair" title="Perko pair">Perko pair</a> (10<sub>161</sub>)</li> <li><a href="/wiki/Conway_knot" title="Conway knot">Conway knot</a> (11n34)</li> <li><a href="/wiki/Kinoshita%E2%80%93Terasaka_knot" title="Kinoshita–Terasaka knot">Kinoshita–Terasaka knot</a> (11n42)</li> <li><a href="/wiki/(%E2%88%922,3,7)_pretzel_knot" title="(−2,3,7) pretzel knot">(−2,3,7) pretzel</a> (12n242)</li> <li><a href="/wiki/Whitehead_link" title="Whitehead link">Whitehead</a> (5<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Borromean_rings" title="Borromean rings">Borromean rings</a> (6<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sub></span></span>)</li> <li><a href="/wiki/L10a140_link" title="L10a140 link">L10a140</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Satellite_knot" title="Satellite knot">Satellite</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composite_knot" class="mw-redirect" title="Composite knot">Composite knots</a> <ul><li><a href="/wiki/Granny_knot_(mathematics)" title="Granny knot (mathematics)">Granny</a></li> <li><a href="/wiki/Square_knot_(mathematics)" title="Square knot (mathematics)">Square</a></li></ul></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Knot sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Torus_knot" title="Torus knot">Torus</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Unknot" title="Unknot">Unknot</a> (0<sub>1</sub>)</li> <li><a href="/wiki/Trefoil_knot" title="Trefoil knot">Trefoil</a> (3<sub>1</sub>)</li> <li><a href="/wiki/Cinquefoil_knot" title="Cinquefoil knot">Cinquefoil</a> (5<sub>1</sub>)</li> <li><a href="/wiki/7%E2%82%81_knot" class="mw-redirect" title="7₁ knot">Septafoil</a> (7<sub>1</sub>)</li> <li><a href="/wiki/Unlink" title="Unlink">Unlink</a> (0<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Hopf_link" title="Hopf link">Hopf</a> (2<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li> <li><a href="/wiki/Solomon%27s_knot" title="Solomon's knot">Solomon's</a> (4<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Knot_invariant" title="Knot invariant">Invariants</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_knot" title="Alternating knot">Alternating</a></li> <li><a href="/wiki/Arf_invariant_of_a_knot" title="Arf invariant of a knot">Arf invariant</a></li> <li><a href="/wiki/Bridge_number" title="Bridge number">Bridge no.</a> <ul><li><a href="/wiki/2-bridge_knot" title="2-bridge knot">2-bridge</a></li></ul></li> <li><a href="/wiki/Brunnian_link" title="Brunnian link">Brunnian</a></li> <li><a href="/wiki/Chiral_knot" title="Chiral knot">Chirality</a> <ul><li><a href="/wiki/Invertible_knot" title="Invertible knot">Invertible</a></li></ul></li> <li><a href="/wiki/Crosscap_number" title="Crosscap number">Crosscap no.</a></li> <li><a href="/wiki/Crossing_number_(knot_theory)" title="Crossing number (knot theory)">Crossing no.</a></li> <li><a href="/wiki/Finite_type_invariant" title="Finite type invariant">Finite type invariant</a></li> <li><a href="/wiki/Hyperbolic_volume" title="Hyperbolic volume">Hyperbolic volume</a></li> <li><a href="/wiki/Khovanov_homology" title="Khovanov homology">Khovanov homology</a></li> <li><a href="/wiki/Knot_genus" class="mw-redirect" title="Knot genus">Genus</a></li> <li><a href="/wiki/Knot_group" title="Knot group">Knot group</a></li> <li><a href="/wiki/Link_group" title="Link group">Link group</a></li> <li><a href="/wiki/Linking_number" title="Linking number">Linking no.</a></li> <li><a href="/wiki/Knot_polynomial" title="Knot polynomial">Polynomial</a> <ul><li><a class="mw-selflink selflink">Alexander</a></li> <li><a href="/wiki/Bracket_polynomial" title="Bracket polynomial">Bracket</a></li> <li><a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones</a></li> <li><a href="/wiki/Kauffman_polynomial" title="Kauffman polynomial">Kauffman</a></li></ul></li> <li><a href="/wiki/Pretzel_link" title="Pretzel link">Pretzel</a></li> <li><a href="/wiki/Prime_knot" title="Prime knot">Prime</a> <ul><li><a href="/wiki/List_of_prime_knots" title="List of prime knots">list</a></li></ul></li> <li><a href="/wiki/Stick_number" title="Stick number">Stick no.</a></li> <li><a href="/wiki/Tricolorability" title="Tricolorability">Tricolorability</a></li> <li><a href="/wiki/Unknotting_number" title="Unknotting number">Unknotting no.</a> and <a href="/wiki/Unknotting_problem" title="Unknotting problem">problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation<br />and <a href="/wiki/Knot_operation" title="Knot operation">operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%E2%80%93Briggs_notation" class="mw-redirect" title="Alexander–Briggs notation">Alexander–Briggs notation</a></li> <li><a href="/wiki/Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation</a></li> <li><a href="/wiki/Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></li> <li><a href="/wiki/Flype" title="Flype">Flype</a></li> <li><a href="/wiki/Mutation_(knot_theory)" title="Mutation (knot theory)">Mutation</a></li> <li><a href="/wiki/Reidemeister_move" title="Reidemeister move">Reidemeister move</a></li> <li><a href="/wiki/Skein_relation" title="Skein relation">Skein relation</a></li> <li><a href="/wiki/Knot_tabulation" title="Knot tabulation">Tabulation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander%27s_theorem" title="Alexander's theorem">Alexander's theorem</a></li> <li><a href="/wiki/Berge_knot" title="Berge knot">Berge</a></li> <li><a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">Braid theory</a></li> <li><a href="/wiki/Conway_sphere" title="Conway sphere">Conway sphere</a></li> <li><a href="/wiki/Knot_complement" title="Knot complement">Complement</a></li> <li><a href="/wiki/Double_torus_knot" class="mw-redirect" title="Double torus knot">Double torus</a></li> <li><a href="/wiki/Fibered_knot" title="Fibered knot">Fibered</a></li> <li><a href="/wiki/Knot" title="Knot">Knot</a></li> <li><a href="/wiki/List_of_mathematical_knots_and_links" title="List of mathematical knots and links">List of knots and links</a></li> <li><a href="/wiki/Ribbon_knot" title="Ribbon knot">Ribbon</a></li> <li><a href="/wiki/Slice_knot" title="Slice knot">Slice</a></li> <li><a href="/wiki/Knot_sum" class="mw-redirect" title="Knot sum">Sum</a></li> <li><a href="/wiki/Tait_conjectures" title="Tait conjectures">Tait conjectures</a></li> <li><a href="/wiki/Twist_knot" title="Twist knot">Twist</a></li> <li><a href="/wiki/Wild_knot" title="Wild knot">Wild</a></li> <li><a href="/wiki/Writhe" title="Writhe">Writhe</a></li> <li><a href="/wiki/Surgery_theory" title="Surgery theory">Surgery theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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