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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="knot_theory">Knot theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/knot">knot</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/link">link</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+complement">knot complement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knot+diagrams">knot diagrams</a>, <a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reidemeister+move">Reidemeister move</a></p> </li> </ul> <p><strong>Examples/classes:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trefoil+knot">trefoil knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/torus+knot">torus knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+knot">singular knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+knot">hyperbolic knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borromean+link">Borromean link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+link">Whitehead link</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+link">Hopf link</a></p> </li> </ul> <p><strong>Types</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+knot">prime knot</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mutant+knot">mutant knot</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/knot+invariants">knot invariants</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/crossing+number">crossing number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bridge+number">bridge number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unknotting+number">unknotting number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colorability">colorability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jones+polynomial">Jones polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HOMFLY+polynomial">HOMFLY polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+polynomial">Alexander polynomial</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+invariants">Reshetikhin-Turaev invariants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+knot+invariants">Vassiliev knot invariants</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+observables">observables</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">pertrubative</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/link+invariants">link invariants</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Milnor+mu-bar+invariants">Milnor mu-bar invariants</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linking+number">linking number</a></p> </li> </ul> <p><strong>Related concepts:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Vassiliev+skein+relation">Vassiliev skein relation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seifert+surface">Seifert surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+knot+theory">virtual knot theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a>, <a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/volume+conjecture">volume conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/knot+theory">knot theory</a></div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>What is called <em>Kirby calculus</em> is a way of identifying/classifying <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a> in terms of <a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a> along <a class="existingWikiWord" href="/nlab/show/framed+links">framed links</a> in the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a>.</p> <h2 id="references">References</h2> <p>The original article is</p> <ul> <li id="Kirby78"><a class="existingWikiWord" href="/nlab/show/Rob+Kirby">Rob Kirby</a>, <em>A Calculus for Framed Links in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math></em>. Inventiones Mathematicae, vol. 45 (1978), pp. 35–56 (<a href="https://eudml.org/doc/142535">dml:142535</a>)</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Kirby_calculus">Kirby calculus</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 16, 2024 at 16:33:31. 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