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class="vector-toc-link" href="#Definition_by_pass_equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition by pass equivalence</span> </div> </a> <ul id="toc-Definition_by_pass_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_by_partition_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_by_partition_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Definition by partition function</span> </div> </a> <ul id="toc-Definition_by_partition_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_by_Alexander_polynomial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_by_Alexander_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Definition by Alexander polynomial</span> </div> </a> <ul id="toc-Definition_by_Alexander_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arf_as_knot_concordance_invariant" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arf_as_knot_concordance_invariant"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Arf as knot concordance invariant</span> </div> </a> <ul id="toc-Arf_as_knot_concordance_invariant-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">6</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 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Knot invariant named after Cahit Arf</div> <p>In the mathematical field of <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, the <b>Arf invariant</b> of a knot, named after <a href="/wiki/Cahit_Arf" title="Cahit Arf">Cahit Arf</a>, is a <a href="/wiki/Knot_invariant" title="Knot invariant">knot invariant</a> obtained from a quadratic form associated to a <a href="/wiki/Seifert_surface" title="Seifert surface">Seifert surface</a>. If <i>F</i> is a Seifert surface of a knot, then the <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology group</a> <span class="nowrap">H<sub>1</sub>(<i>F</i>, <b>Z</b>/2<b>Z</b>)</span> has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The <a href="/wiki/Arf_invariant" title="Arf invariant">Arf invariant</a> of this quadratic form is the Arf invariant of the knot. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_by_Seifert_matrix">Definition by Seifert matrix</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arf_invariant_of_a_knot&action=edit&section=1" title="Edit section: Definition by Seifert matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 V=v_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=v_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6e2969b838ae3f83e2ac082fd799ee0c34c7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.948ex; height:2.843ex;" alt="{\displaystyle V=v_{i,j}}"></span> be a <a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert matrix</a> of the knot, constructed from a set of curves on a <a href="/wiki/Seifert_surface" title="Seifert surface">Seifert surface</a> of genus <i>g</i> which represent a basis for the first <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> of the surface. This means that <i>V</i> is a <span class="nowrap">2<i>g</i> × 2<i>g</i></span> matrix with the property that <span class="nowrap"><i>V</i> − <i>V</i><sup>T</sup></span> is a <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrix</a>. The <i>Arf invariant</i> of the knot is the residue of </p> <dl><dd><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 \sum \limits _{i=1}^{g}v_{2i-1,2i-1}v_{2i,2i}{\pmod {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munderover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>,</mo> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum \limits _{i=1}^{g}v_{2i-1,2i-1}v_{2i,2i}{\pmod {2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60777e70ddf6f91209549d8bb44d3b5b2e60f81e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.629ex; height:7.009ex;" alt="{\displaystyle \sum \limits _{i=1}^{g}v_{2i-1,2i-1}v_{2i,2i}{\pmod {2}}.}"></span></dd></dl> <p>Specifically, 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 \{a_{i},b_{i}\},i=1\ldots g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>…</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{i},b_{i}\},i=1\ldots g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96f5e9144770a4811c237685fe8918b68172632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.896ex; height:2.843ex;" alt="{\displaystyle \{a_{i},b_{i}\},i=1\ldots g}"></span>, is a symplectic basis for the intersection form on the Seifert surface, then </p> <dl><dd><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 \operatorname {Arf} (K)=\sum \limits _{i=1}^{g}\operatorname {lk} \left(a_{i},a_{i}^{+}\right)\operatorname {lk} \left(b_{i},b_{i}^{+}\right){\pmod {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Arf</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munderover> <mi>lk</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>lk</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Arf} (K)=\sum \limits _{i=1}^{g}\operatorname {lk} \left(a_{i},a_{i}^{+}\right)\operatorname {lk} \left(b_{i},b_{i}^{+}\right){\pmod {2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743540b1a32244cb269b8533576a7c7524f312ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.115ex; height:7.009ex;" alt="{\displaystyle \operatorname {Arf} (K)=\sum \limits _{i=1}^{g}\operatorname {lk} \left(a_{i},a_{i}^{+}\right)\operatorname {lk} \left(b_{i},b_{i}^{+}\right){\pmod {2}}.}"></span></dd></dl> <p>where lk is the <a href="/wiki/Link_number" class="mw-redirect" title="Link number">link number</a> and <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^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe313cdd96c9f3af676e65d62be8a89618653a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.741ex; height:2.509ex;" alt="{\displaystyle a^{+}}"></span> denotes the positive pushoff of <i>a</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_by_pass_equivalence">Definition by pass equivalence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arf_invariant_of_a_knot&action=edit&section=2" title="Edit section: Definition by pass equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This approach to the Arf invariant is due to <a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Louis Kauffman</a>. </p><p>We define two knots to be <b>pass equivalent</b> if they are related by a finite sequence of pass-moves.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Every knot is pass-equivalent to either the <a href="/wiki/Unknot" title="Unknot">unknot</a> or the <a href="/wiki/Trefoil_knot" title="Trefoil knot">trefoil</a>; these two knots are not pass-equivalent and additionally, the right- and left-handed trefoils are pass-equivalent.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Now we can define the Arf invariant of a knot to be 0 if it is pass-equivalent to the unknot, or 1 if it is pass-equivalent to the trefoil. This definition is equivalent to the one above. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_by_partition_function">Definition by partition function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arf_invariant_of_a_knot&action=edit&section=3" title="Edit section: Definition by partition function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Vaughan_Jones" title="Vaughan Jones">Vaughan Jones</a> showed that the Arf invariant can be obtained by taking the partition function of a signed planar graph associated to a <a href="/wiki/Knot_diagram" class="mw-redirect" title="Knot diagram">knot diagram</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_by_Alexander_polynomial">Definition by Alexander polynomial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arf_invariant_of_a_knot&action=edit&section=4" title="Edit section: Definition by Alexander polynomial"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This approach to the Arf invariant is by Raymond Robertello.<sup id="cite_ref-Robertello_3-0" class="reference"><a href="#cite_note-Robertello-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Let </p> <dl><dd><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 (t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b46cb76f46439b50cf27728ec5e41684dcc803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.518ex; height:3.176ex;" alt="{\displaystyle \Delta (t)=c_{0}+c_{1}t+\cdots +c_{n}t^{n}+\cdots +c_{0}t^{2n}}"></span></dd></dl> <p>be the <a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a> of the knot. Then the Arf invariant is the residue of </p> <dl><dd><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_{n-1}+c_{n-3}+\cdots +c_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n-1}+c_{n-3}+\cdots +c_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f4f7f30c5c45ef318308677d225701fbb19c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.877ex; height:2.343ex;" alt="{\displaystyle c_{n-1}+c_{n-3}+\cdots +c_{r}}"></span></dd></dl> <p>modulo 2, where <span class="nowrap"><i>r</i> = 0</span> for <i>n</i> odd, and <span class="nowrap"><i>r</i> = 1</span> for <i>n</i> even. </p><p>Kunio Murasugi<sup id="cite_ref-Murasugi_4-0" class="reference"><a href="#cite_note-Murasugi-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> proved that the Arf invariant is zero if and only if <span class="nowrap">Δ(−1) ≡ ±1 modulo 8</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Arf_as_knot_concordance_invariant">Arf as knot concordance invariant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arf_invariant_of_a_knot&action=edit&section=5" title="Edit section: Arf as knot concordance invariant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the Fox-Milnor criterion, which tells us that the Alexander polynomial of a <a href="/wiki/Slice_knot" title="Slice knot">slice knot</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 K\subset \mathbb {S} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset \mathbb {S} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0af63e077237c7e1d23bcdf007a69ccde62a98b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.511ex; height:2.676ex;" alt="{\displaystyle K\subset \mathbb {S} ^{3}}"></span> factors as <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 (t)=p(t)p\left(t^{-1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mrow> <mo>(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (t)=p(t)p\left(t^{-1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4169610cfd156ee82ef7c06815d51c9f28bd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.36ex; height:3.343ex;" alt="{\displaystyle \Delta (t)=p(t)p\left(t^{-1}\right)}"></span> for some 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 p(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b827c545ca1487214f0c498131228ef87718ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle p(t)}"></span> with integer coefficients, we know that the determinant <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 \left|\Delta (-1)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\Delta (-1)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371c943ff48b7d96a91f58f0aef9147e0349899a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.009ex; height:2.843ex;" alt="{\displaystyle \left|\Delta (-1)\right|}"></span> of a slice knot is a square integer. As <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 \left|\Delta (-1)\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\Delta (-1)\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371c943ff48b7d96a91f58f0aef9147e0349899a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.009ex; height:2.843ex;" alt="{\displaystyle \left|\Delta (-1)\right|}"></span> is an odd integer, it has to be congruent to 1 modulo 8. Combined with Murasugi's result, this shows that the Arf invariant of a slice knot vanishes. </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=Arf_invariant_of_a_knot&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Kauffman (1987) p.74</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Kauffman (1987) pp.75–78</span> </li> <li id="cite_note-Robertello-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Robertello_3-0">^</a></b></span> <span class="reference-text">Robertello, Raymond, An Invariant of Knot Corbordism, <a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" title="Communications on Pure and Applied Mathematics">Communications on Pure and Applied Mathematics</a>, Volume 18, pp. 543–555, 1965</span> </li> <li id="cite_note-Murasugi-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Murasugi_4-0">^</a></b></span> <span class="reference-text">Murasugi, Kunio, The Arf Invariant for Knot Types, Proceedings of the American Mathematical Society, Vol. 21, No. 1. (Apr., 1969), pp. 69–72</span> </li> </ol></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=Arf_invariant_of_a_knot&action=edit&section=7" title="Edit section: References"><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="CITEREFKauffman1983" class="citation book cs1"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis H.</a> (1983). <i>Formal knot theory</i>. Mathematical notes. Vol. 30. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08336-3" title="Special:BookSources/0-691-08336-3"><bdi>0-691-08336-3</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.series=Mathematical+notes&rft.pub=Princeton+University+Press&rft.date=1983&rft.isbn=0-691-08336-3&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArf+invariant+of+a+knot" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKauffman1987" class="citation book cs1"><a href="/wiki/Louis_Kauffman" title="Louis Kauffman">Kauffman, Louis H.</a> (1987). <i>On knots</i>. Annals of Mathematics Studies. Vol. 115. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08435-1" title="Special:BookSources/0-691-08435-1"><bdi>0-691-08435-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+knots&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=1987&rft.isbn=0-691-08435-1&rft.aulast=Kauffman&rft.aufirst=Louis+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArf+invariant+of+a+knot" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirby1989" class="citation book cs1"><a href="/wiki/Robion_Kirby" title="Robion Kirby">Kirby, Robion</a> (1989). <i>The topology of 4-manifolds</i>. Lecture Notes in Mathematics. Vol. 1374. <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-51148-2" title="Special:BookSources/0-387-51148-2"><bdi>0-387-51148-2</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+topology+of+4-manifolds&rft.series=Lecture+Notes+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1989&rft.isbn=0-387-51148-2&rft.aulast=Kirby&rft.aufirst=Robion&rfr_id=info%3Asid%2Fen.wikipedia.org%3AArf+invariant+of+a+knot" class="Z3988"></span></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 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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)" 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 class="mw-selflink selflink">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 href="/wiki/Alexander_polynomial" title="Alexander polynomial">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 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