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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 #3 to #4: <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> <h4 id='riemannian_geometry'>Riemannian geometry</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a></strong></p> <h2 id='basic_definitions'>Basic definitions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Riemannian+manifold'>Riemannian manifold</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric</a>, <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/isometry+group'>isometry group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+Riemannian+metrics'>moduli space of Riemannian metrics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+manifold'>pseudo-Riemannian manifold</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/smooth+Lorentzian+space'>Lorentzian manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic'>geodesic</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic+convexity'>geodesic convexity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geodesic+flow'>geodesic flow</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Levi-Civita+connection'>Levi-Civita connection</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Riemann+curvature'>Riemann curvature</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+metric+connection'>torsion of a metric connection</a></li> </ul> </li> </ul> <h2 id='further_concepts'>Further concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+star+operator'>Hodge inner product</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hodge+star+operator'>Hodge star operator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gradient'>gradient</a>, <a class='existingWikiWord' href='/nlab/show/diff/gradient+flow'>gradient flow</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+conjecture'>Poincaré conjecture</a>-theorem</li> </ul> <h2 id='applications'>Applications</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gravity'>gravity</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Einstein-Hilbert+action'>Einstein-Hilbert action</a>, <a class='existingWikiWord' href='/nlab/show/diff/Einstein+equation'>Einstein equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a></li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/Riemannian+geometry+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a> <math class='maruku-mathml' display='inline' id='mathml_9ed810e25697185c24600c75bdee8702dc80f831_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>⊂</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>L \subset S^3</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/diff/3-sphere'>3-sphere</a> is called a <em>hyperbolic link</em> if its <a class='existingWikiWord' href='/nlab/show/diff/knot+complement'>knot complement</a> <math class='maruku-mathml' display='inline' id='mathml_9ed810e25697185c24600c75bdee8702dc80f831_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>∖</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>S^3 \setminus L</annotation></semantics></math> admits the <a class='existingWikiWord' href='/nlab/show/diff/structure'>structure</a> of a <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+geometry'>hyperbolic metric</a>, hence of a <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+3-manifold'>hyperbolic 3-manifold</a>, hence if it is <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometric</a> to a <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> of <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+3-space'>hyperbolic 3-space</a> by a <a class='existingWikiWord' href='/nlab/show/diff/torsion+subgroup'>torsion-free</a> <a class='existingWikiWord' href='/nlab/show/diff/discrete+group'>discrete group</a> <math class='maruku-mathml' display='inline' id='mathml_9ed810e25697185c24600c75bdee8702dc80f831_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/action'>acting</a> by <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometries</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9ed810e25697185c24600c75bdee8702dc80f831_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>∖</mo><mi>L</mi><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msup><mi>ℍ</mi> <mn>3</mn></msup><mo stretchy='false'>/</mo><mi>Γ</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> S^3 \setminus L \;\simeq\; \mathbb{H}^3/\Gamma \,. </annotation></semantics></math></div> <p>(e.g. <a href='#FKP17'>FKP 17, Def. 2.8</a>)</p> <p>If the hyperbolic link is in fact a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a> (has a single <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected component</a>) it is called a <em>hyperbolic knot</em>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+3-manifold'>hyperbolic 3-manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/volume+conjecture'>volume conjecture</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li id='FKP17'>David Futer, Efstratia Kalfagianni, Jessica S. Purcell, <em>A survey of hyperbolic knot theory</em><span><del class='diffmod'> </del><ins class='diffmod'> ,</ins><ins class='diffins'> Springer</ins><ins class='diffins'> Proceedings</ins><ins class='diffins'> in</ins><ins class='diffins'> Mathematics</ins><ins class='diffins'> &</ins><ins class='diffins'> Statistics,</ins><ins class='diffins'> vol.</ins><ins class='diffins'> 284</ins><ins class='diffins'> (2019),</ins><ins class='diffins'> 1-30</ins> (</span><a href='https://arxiv.org/abs/1708.07201'>arXiv:1708.07201</a>)</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Hyperbolic_link'>Hyperbolic link</a></em></li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 12, 2019 at 07:41:22. 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