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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #9 to #10: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='knot_theory'>Knot theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot'>knot theory</a></strong></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a></strong>, <strong><a class='existingWikiWord' href='/nlab/show/diff/link'>link</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/knot+complement'>knot complement</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagrams</a>, <a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>chord diagram</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister move</a></p> </li> </ul> <p><strong>Examples/classes:</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trefoil+knot'>trefoil knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/torus+knot'>torus knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/singular+knot'>singular knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+link'>hyperbolic knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Borromean+link'>Borromean link</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+link'>Whitehead link</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hopf+link'>Hopf link</a></p> </li> </ul> <p><strong>Types</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/prime+knot'>prime knot</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mutant+knot'>mutant knot</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>knot invariants</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossing+number'>crossing number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bridge+number'>bridge number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unknotting+number'>unknotting number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colorable+knot'>colorability</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/knot+group'>knot group</a></p> </li> <li> <p><span class='newWikiWord'>knot genus<a href='/nlab/new/knot+genus'>?</a></span></p> </li> <li> <p>polynomial knot invariants</p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/non-perturbative+quantum+field+theory'>non-perturbative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>)</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/HOMFLY-PT+polynomial'>HOMFLY polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Alexander+polynomial'>Alexander polynomial</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reshetikhin-Turaev+construction'>Reshetikhin-Turaev invariants</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+invariant'>Vassiliev knot invariants</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>pertrubative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Khovanov+homology'>Khovanov homology</a></p> </li> <li> <p><span class='newWikiWord'>Kauffman bracket<a href='/nlab/new/Kauffman+bracket'>?</a></span></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/link+invariant'>link invariants</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Milnor+mu-bar+invariant'>Milnor mu-bar invariants</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/linking+number'>linking number</a></p> </li> </ul> <p><strong>Related concepts:</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+skein+relation'>Vassiliev skein relation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Seifert+surface'>Seifert surface</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/virtual+knot+theory'>virtual knot theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dehn+surgery'>Dehn surgery</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kirby+calculus'>Kirby calculus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/volume+conjecture'>volume conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/arithmetic+topology'>arithmetic topology</a></p> </li> </ul> </div> </div> </div> <h1 id='the_homflypt_polynomial'>The HOMFLY-PT Polynomial</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><ins class='diffins'><li><a href='#related_entries'>Related entries</a></li></ins><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The HOMFLY-PT polynomial is a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a> and <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a> <a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>invariant</a>. Confusingly, there are several variants depending on exactly which relationships are used to define it. All are related by simple substitutions.</p> <h2 id='definition'>Definition</h2> <p>To compute the HOMFLY-PT polynomial, one starts from an <a class='existingWikiWord' href='/nlab/show/diff/oriented+link+diagram'>oriented link diagram</a> and uses the following rules:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is an isotopy invariant (thus, unchanged by <a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister moves</a>).</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo stretchy='false'>(</mo><mtext>unknot</mtext><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>P(\text{unknot}) = 1</annotation></semantics></math></p> </li> <li> <p>Let <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>L_+</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mo>−</mo></msub></mrow><annotation encoding='application/x-tex'>L_-</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>L_0</annotation></semantics></math> be links which are the same except for one part where they differ according to the diagrams below. Then, depending on the choice of variables:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>l</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo>+</mo><msup><mi>l</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo><mo>+</mo><mi>m</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0</annotation></semantics></math>.</li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo>−</mo><msup><mi>a</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>z</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a \cdot P(L_+) - a^{-1} \cdot P(L_-) = z \cdot P(L_0)</annotation></semantics></math>. (Sometimes <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ν</mi></mrow><annotation encoding='application/x-tex'>\nu</annotation></semantics></math> is used instead of <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>)</li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>α</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo>−</mo><mi>α</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>z</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0)</annotation></semantics></math>.</li> <li>Using <em>three</em> variables: <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>+</mo></msub><mo stretchy='false'>)</mo><mo>+</mo><mi>y</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mo>−</mo></msub><mo stretchy='false'>)</mo><mo>+</mo><mi>z</mi><mo>⋅</mo><mi>P</mi><mo stretchy='false'>(</mo><msub><mi>L</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0</annotation></semantics></math>.</li> </ol> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mtable columnalign='center center center' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><semantics><annotation-xml encoding='SVG1.1'><svg height='62pt' viewBox='-2.5 -2.5 61.90549 61.90549 ' width='62pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'><g transform='translate(0 59) scale(1 -1) translate(0 2.5)'><g stroke='#000'><g fill='#000'><g stroke-width='.4pt'><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff'><g stroke-width='5pt'><path d='m57 0l-57 57' fill='none' /><g stroke-width='1pt'><g stroke='#f00'><path d='m57 0l-57 57' fill='none' /></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(-.71 .71 -.71 -.71 1.7 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' 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59) scale(1 -1) translate(0 2.5)'><g stroke='#000'><g fill='#000'><g stroke-width='.4pt'><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff'><g stroke-width='5pt'><path d='m0 0l57 57' fill='none' /><g stroke-width='1pt'><g stroke='#f00'><path d='m0 0l57 57' fill='none' /></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(.71 .71 -.71 .71 55 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff' /></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff'><g stroke-width='5pt'><path d='m57 0l-57 57' fill='none' /><g stroke-width='1pt'><g stroke='#f00'><path d='m57 0l-57 57' fill='none' /></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(-.71 .71 -.71 -.71 1.7 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff' /></g></g></g></g></g></g></g></g></g></svg></annotation-xml></semantics></mtd> <mtd><semantics><annotation-xml encoding='SVG1.1'><svg height='62pt' viewBox='-2.5 -2.5 61.90549 61.90549 ' width='62pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'><g transform='translate(0 59) scale(1 -1) translate(0 2.5)'><g stroke='#000'><g fill='#000'><g stroke-width='.4pt'><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff'><g stroke-width='5pt'><path d='m0 0c21 21 21 36 .28 57' fill='none' /><g stroke-width='1pt'><g stroke='#f00'><path d='m0 0c21 21 21 36 .28 57' fill='none' /></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(-.71 .71 -.71 -.71 2 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(-.71 .71 -.71 -.71 2 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff' /></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff'><g stroke-width='5pt'><path d='m57 0c-21 21-21 36 0 57' fill='none' /><g stroke-width='1pt'><g stroke='#f00'><path d='m57 0c-21 21-21 36 0 57' fill='none' /></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(.71 .71 -.71 .71 55 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke='#f00'><g fill='#f00'><g transform='matrix(.71 .71 -.71 .71 55 55)'><g transform='matrix(1 0 0 1 0 0)'><g stroke-dasharray='none' stroke-dashoffset='0pt'><g stroke-linejoin='miter'><path d='m-7.3 4.5l7.8-4.5-7.8-4.5z' /></g></g></g></g></g></g><g stroke-width='2pt'><g stroke='#fff'><g fill='#fff' /></g></g></g></g></g></g></g></g></g></svg></annotation-xml></semantics></mtd></mtr> <mtr><mtd><msub><mi>L</mi> <mo>+</mo></msub></mtd> <mtd><msub><mi>L</mi> <mo>−</mo></msub></mtd> <mtd><msub><mi>L</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><annotation encoding='application/x-tex'> \begin{array}{ccc} \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0c21 21 21 36 .28 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0c21 21 21 36 .28 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 2 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 2 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0c-21 21-21 36 0 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0c-21 21-21 36 0 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} \\ L_+ & L_- & L_0 \end{array} </annotation></semantics></math></div></li> </ol> <p>From the rules, one can read off the relationships between the different formulations:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>=</mo><mi>α</mi><mo>=</mo><msup><mi>a</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>y = \alpha = a^{-1}</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mi>α</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>x = - \alpha^{-1} = -a</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>i</mi><mi>l</mi></mrow><annotation encoding='application/x-tex'>a = - i l</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>l</mi><mo>=</mo><mi>i</mi><mi>a</mi></mrow><annotation encoding='application/x-tex'>l = i a</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><mi>i</mi><mi>m</mi></mrow><annotation encoding='application/x-tex'>z = i m</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>i</mi><mi>z</mi></mrow><annotation encoding='application/x-tex'>m = - i z</annotation></semantics></math>.</li> </ol> <h2 id='properties'>Properties</h2> <p>The HOMFLY polynomial generalises both the <a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a> and the <a class='existingWikiWord' href='/nlab/show/diff/Alexander+polynomial'>Alexander polynomial</a> (equivalently, the <a class='existingWikiWord' href='/nlab/show/diff/Conway+polynomial'>Conway polynomial</a>).</p> <ul> <li> <p>To get the <a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a>, make one of the following substitutions:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>a = q^{-1}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>z = q^{1/2} - q^{-1/2}</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>=</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>\alpha = q</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>z = q^{1/2} - q^{-1/2}</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>l</mi><mo>=</mo><mi>i</mi><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>l = i q^{-1}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>=</mo><mi>i</mi><mo stretchy='false'>(</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>m = i (q^{-1/2} - q^{1/2})</annotation></semantics></math></li> </ol> </li> <li> <p>To get the <a class='existingWikiWord' href='/nlab/show/diff/Conway+polynomial'>Conway polynomial</a>, make one of the following substitutions:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>a = 1</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\alpha = 1</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>l</mi><mo>=</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>l = i</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>i</mi><mi>z</mi></mrow><annotation encoding='application/x-tex'>m = -i z</annotation></semantics></math></li> </ol> </li> <li> <p>To get the <a class='existingWikiWord' href='/nlab/show/diff/Alexander+polynomial'>Alexander polynomial</a>, make one of the following substitutions:</p> <ol> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>a = 1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>z = q^{1/2} - q^{-1/2}</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\alpha = 1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>z = q^{1/2} - q^{-1/2}</annotation></semantics></math></li> <li><math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>l</mi><mo>=</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>l = i</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_073549ae0f879a33ae8b9a392cd005168d733595_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>=</mo><mi>i</mi><mo stretchy='false'>(</mo><msup><mi>q</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn></mrow></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>m = i (q^{-1/2} - q^{1/2})</annotation></semantics></math></li> </ol> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><h2 id='related_entries'>Related entries</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Shum%27s+theorem'>Shum's theorem</a></li> </ul></ins> <h2 id='references'>References</h2> <p>See the <a href='http://en.wikipedia.org/wiki/HOMFLY_polynomial'>wikipedia page</a> for the origin of the name.</p> <p>Some fairly elementary discussion of the HOMFLY polynomial is given in introductory texts such as</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Nick+Gilbert'>N. D. Gilbert</a> and <a class='existingWikiWord' href='/nlab/show/diff/Tim+Porter'>T. Porter</a>, Knots and Surfaces, Oxford U.P., 1994.</li> </ul> <p>The original work was published as</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Peter+Freyd'>P. Freyd</a>, <a class='existingWikiWord' href='/nlab/show/diff/David+Yetter'>D. Yetter</a>, J. Hoste, W.B.R. Lickorish, K. Millett, and <a class='existingWikiWord' href='/nlab/show/diff/Adrian+Ocneanu'>A. Ocneanu</a>. (1985). <em>A New Polynomial Invariant of Knots and Links</em> Bulletin of the American Mathematical Society 12 (2): 239–246.</li> </ul> <p>More recent work includes:</p> <ul> <li>A.Mironov, A.Morozov, An.Morozov, <em>Character expansion for HOMFLY polynomials. I. Integrability and difference equations</em>, <a href='http://arxiv.org/abs/1112.5754'>arxiv/1112.5754</a></li> <li>Hugh Morton, Peter Samuelson, <em>The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra</em>, <a href='http://arxiv.org/abs/1410.0859'>arxiv/1410.0859</a></li> </ul> <p><div class='property'> category: <a class='category_link' href='/nlab/list/knot+theory'>knot theory</a></div></p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 31, 2024 at 18:38:59. 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