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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 #10 to #11: <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='#formal_definition'>Formal definition</a></li><li><a href='#obtaining_a_link_diagram_from_a_link'>Obtaining a link diagram from a link</a></li><li><a href='#related_concepts'>Related concepts</a></li><ins class='diffins'><li><a href='#references'>References</a></li></ins></ul></div> <h2 id='idea'>Idea</h2> <p>A <em>link diagram</em> is, roughly speaking, the combinatorial object obtained by projecting a <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a> in ‘general position’ to a plane. <a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister's theorem</a> establishes that one does not lose any essential information by passing from a link to its diagram, and thus it is possible to study knot theory in a way which takes link diagrams as primary: this is sometimes known as <a class='existingWikiWord' href='/nlab/show/diff/diagrammatic+knot+theory'>diagrammatic knot theory</a>.</p> <h2 id='formal_definition'>Formal definition</h2> <p>The formal definition of a link diagram is in itself independent of the notion of a <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a>.</p> <p>\begin{defn} \label{DefinitionConnectedLinkDiagram} A <em>connected link diagram</em> is a <a class='existingWikiWord' href='/nlab/show/diff/connected+graph'>connected</a> (undirected) 4-valent <a class='existingWikiWord' href='/nlab/show/diff/plane+graph'>plane graph</a> <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> equipped with the following data for every vertex <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <ol> <li> <p>A choice of division of the four edges incident to <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> into two pairs, say <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>e</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(e_{0}, e_{1})</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>e</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>3</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(e_{2}, e_{3})</annotation></semantics></math>. We refer to the two edges in the first pair as <em>over-edges</em>, and to the two edges <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(e_{1}, e_{2})</annotation></semantics></math> in the second pair as <em>under-edges</em>.</p> </li> <li> <p>A cyclic ordering of the four edges of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> incident to <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> so that the over-edges and under-edges alternate.</p> </li> </ol> <p>\end{defn}</p> <p>\begin{defn} \label{DefinitionLinkDiagram} A <em>link diagram</em> is a planar graph such that each connected component satisfies either 1. or 2. below.</p> <ol> <li> <p>It is (up to homeomorphism) a circle, disjoint from the rest of the link diagram.</p> </li> <li> <p>It is a connected link diagram in the sense of Definition \ref{DefinitionConnectedLinkDiagram}.</p> </li> </ol> <p>\end{defn}</p> <p>\begin{rmk} It is important to note that a plane graph consists, by definition, of an abstract <a class='existingWikiWord' href='/nlab/show/diff/graph'>graph</a> together with a <em>chosen</em> embedding into the plane. There exist non-equivalent link diagrams which have the same underlying abstract graph, but for which the embedding in the plane is different. \end{rmk}</p> <p>\begin{terminology} A component of a link diagram which satisfies 1. in Definition \ref{DefinitionLinkDiagram} is typically referred to as an <em>unknot</em>, or an <em>unknotted component</em>. \end{terminology}</p> <p>\begin{terminology} \label{TerminologyCrossingsOfALinkDiagram} The vertices <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> of a link diagram <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> which do not belong to an unknotted component, together with the data of 1. and 2. in Definition \ref{DefinitionConnectedLinkDiagram} for <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math>, are referred to as the <em>crossings</em> of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>.</p> <p>\end{terminology}</p> <h2 id='obtaining_a_link_diagram_from_a_link'>Obtaining a link diagram from a link</h2> <p>A link diagram can be obtained from a <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a> by choosing a plane in <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^{3}</annotation></semantics></math>, and projecting the link onto this plane. A small change in the direction of projection will ensure that it is one-to-one except at the double points, which will become the crossings of the link diagram in the sense of Terminology \ref{TerminologyCrossingsOfALinkDiagram}, where the image curve of the knot crosses itself once transversely. Which strand of the two intersecting at the double point is the ‘over strand’ and which is the ‘under strand’ is recorded as the data needed for 1. and 2. in Definition \ref{DefinitionConnectedLinkDiagram}.</p> <p>Consider for example the parallel projection</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>p: \mathbb{R}^3 \to \mathbb{R}^2</annotation></semantics></math></div> <p>defined by <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p(x,y,z) = (x,y,0)</annotation></semantics></math>.</p> <p>(If you prefer your knots to be in <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>S^3</annotation></semantics></math>, of course, you can remove a single point from the complement of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> and then project down to <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mstyle mathvariant='bold'><mi>R</mi></mstyle> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbf{R}^3</annotation></semantics></math>. It does not matter which point you use.)</p> <p>A point <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>x</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{x}</annotation></semantics></math> in the image <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>pK</mi></mrow><annotation encoding='application/x-tex'>pK</annotation></semantics></math> is called a <em>multiple point</em> if <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>p</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>x</mi></mstyle><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p^{-1}(\mathbf{x})</annotation></semantics></math> contains more than one point of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>. A <em>double point</em> occurs when there are exactly two points of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> in this, and similarly for a <em>triple point</em>, etc. Multiple points of infinite order could occur.</p> <p>A knot is in <em>regular position</em> or <em>general position</em> with respect to <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> if there are only double points and these are genuine crossings (i.e. no tangential touching occurs in the projected curve).</p> <p>Any smooth or PL link <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is equivalent under an arbitrarily small rotation of <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mstyle mathvariant='bold'><mi>R</mi></mstyle> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbf{R}^3</annotation></semantics></math> to one in regular position with respect to <math class='maruku-mathml' display='inline' id='mathml_bf291dc80590f8223942b0e1a112a1d6c5d6478f_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>.</p> <p>A proof can be found in Crowell and Fox (page 7).</p> <p>\begin{terminology} A <em>knot diagram</em> is a link diagram which arises from a projection of a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a>.\end{terminology}</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister move</a></li> </ul><ins class='diffins'> </ins><ins class='diffins'><h2 id='references'>References</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p>Sam Nelson: <em>The Combinatorial Revolution in Knot Theory</em>, Notices of the AMS <strong>58</strong> 11 (2011) [[pdf](https://math.mit.edu/~ormsby/nelson-combinatorial-revolution.pdf), full issue: <a href='https://www.ams.org/notices/201111/201111FullIssue.pdf'>pdf</a>]</p> </li> <li> <p>Matt Mastin: <em>Links and Planar Diagram Codes</em> [[arXiv:1309.3288](https://arxiv.org/abs/1309.3288)]</p> </li> </ul></ins> <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 15:50:11. 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