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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 #7 to #8: <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='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_3d_chernsimons_theory'>Relation to 3d Chern-Simons theory</a></li><li><a href='#relation_to_4d_super_yangmills_theory'>Relation to 4d super Yang-Mills theory</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#ReferencesJonesPolynomialAsWilsonLoopObservables'>Jones polynomial as Wilson loop observables</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <em>Jones polynomial</em> is a <a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>knot invariant</a>. It is a special case of the <a class='existingWikiWord' href='/nlab/show/diff/HOMFLY-PT+polynomial'>HOMFLY-PT polynomial</a>. See there for more details.</p> <h2 id='properties'>Properties</h2> <h3 id='relation_to_3d_chernsimons_theory'>Relation to 3d Chern-Simons theory</h3> <p>In (<a href='#Witten89'>Witten 89</a>) it was shown that the Jones polynomial as a polynomial in <math class='maruku-mathml' display='inline' id='mathml_efe39a80d11d40e461c96fe87c74e016f9b545f0_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> is equivalently the <a class='existingWikiWord' href='/nlab/show/diff/partition+function'>partition function</a> of <math class='maruku-mathml' display='inline' id='mathml_efe39a80d11d40e461c96fe87c74e016f9b545f0_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SU</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SU(2)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> with a <a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson loop</a> specified by the given <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a> as a function of the exponentiated <a class='existingWikiWord' href='/nlab/show/diff/level+%28Chern-Simons+theory%29'>level</a> of the Chern-Simons theory. Extensive lecture notes on this are in (<a href='#Witten13a'>Witten 13a</a>).</p> <h3 id='relation_to_4d_super_yangmills_theory'>Relation to 4d super Yang-Mills theory</h3> <p>Later in (<a href='#Witten11'>Witten 11</a>) this identification was further refined to a correspondence between <a class='existingWikiWord' href='/nlab/show/diff/Khovanov+homology'>Khovanov homology</a> and observables in 4-dimensional <a class='existingWikiWord' href='/nlab/show/diff/super+Yang-Mills+theory'>super Yang-Mills theory</a>. Extensive lectures notes on this are in (<a href='#Witten13b'>Witten 13b</a>).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/colored+Jones+polynomial'>colored Jones polynomial</a></li> </ul> <h2 id='references'>References</h2> <h3 id='general'>General</h3> <p>See also:</p> <ul> <li>Wikipedia, <em><a href='https://en.m.wikipedia.org/wiki/Jones_polynomial'>Jones polynomial</a></em></li> </ul><ins class='diffins'> </ins><ins class='diffins'><p>An <a class='existingWikiWord' href='/nlab/show/diff/algorithm'>algorithm</a> for computing the Jones polynomial on a <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological quantum computer</a> based on <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>anyon statistics</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Samuel+J.+Lomonaco'>Samuel J. Lomonaco</a>, <a class='existingWikiWord' href='/nlab/show/diff/Louis+Kauffman'>Louis Kauffman</a>, <em>Topological Quantum Computing and the Jones Polynomial</em>, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) (<a href='https://arxiv.org/abs/quant-ph/0605004'>arXiv:quant-ph/0605004</a>)</li> </ul></ins> <h3 id='ReferencesJonesPolynomialAsWilsonLoopObservables'>Jones polynomial as Wilson loop observables</h3> <p>The identification of the <a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a> with <a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson loop observables</a> in <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a> is due to</p> <ul> <li id='Witten89'><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>Quantum field theory and the Jones polynomial</em>, Commun. Math. Phys. 121,351-399 (1989) (<a href='https://projecteuclid.org/euclid.cmp/1104178138'>euclid.cmp/1104178138</a>)</li> </ul> <p>see also</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/J%C3%BCrg+Fr%C3%B6hlich'>J眉rg Fr枚hlich</a>, C. King, <em>The Chern-Simons theory and knot polynomials</em>, Comm. Math. Phys. Volume 126, Number 1 (1989), 167-199 (<a href='https://projecteuclid.org/euclid.cmp/1104179728'>euclid.cmp/1104179728</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li id='Witten13a'><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>A New Look At The Jones Polynomial of a Knot</em>, Clay Conference, Oxford, October 1, 2013 (<a class='existingWikiWord' href='/nlab/files/WittenOnJonesPolynomial2013.pdf' title='pdf'>pdf</a>)</li> </ul> <p>The <a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>categorification</a> of this relation to an identification of <a class='existingWikiWord' href='/nlab/show/diff/Khovanov+homology'>Khovanov homology</a> with <a class='existingWikiWord' href='/nlab/show/diff/observable'>observables</a> in <a class='existingWikiWord' href='/nlab/show/diff/4d+superconformal+gauge+field+theory'>D=4 super Yang-Mills theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>Khovanov homology and gauge theory</em>, <a href='http://arxiv.org/abs/1108.3103'>arxiv/1108.3103</a></p> </li> <li id='Witten11'> <p><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>Fivebranes and Knots</em> (<a href='http://arxiv.org/abs/1101.3216'>arXiv:1101.3216</a>)</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id='Witten13b'><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>Khovanov Homology And Gauge Theory</em>, Clay Conference, Oxford, October 2, 2013 (<a class='existingWikiWord' href='/nlab/files/WittenOnKhovanovHomology2013.pdf' title='pdf'>pdf</a>)</li> </ul> <p>See also</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Vaughan+Jones'>Vaughan Jones</a>, <em>Index for subfactors</em>, Invent. Math. <strong>72</strong>, I (I983); <em>A polynomial invariant for links via yon Neumann algebras</em>, Bull. AMS <strong>12</strong>, 103 (1985); <em>Hecke algebra representations of braid groups and link polynomials</em>, Ann. Math. <strong>126</strong>, 335 (1987)</li> </ul> <p><div class='property'> category: <a class='category_link' href='/nlab/list/knot+theory'>knot theory</a></div></p> </div> <div class="revisedby"> <p> Last revised on May 21, 2021 at 06:56:32. 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