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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 #1 to #2: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <del class='diffmod'><p>Let <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> be a knot.</p></del><ins class='diffmod'><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></ins> <del class='diffmod'><div class='un_defn'> <h6 id='definition'>Definition</h6> <p>The <em>crossing number</em>, <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c(K)</annotation></semantics></math>, of <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> is the minimum number of crossings in a diagram in the isotopy class of <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>.</p> </div></del><ins class='diffmod'><p>\tableofcontents</p></ins> <del class='diffmod'><p>The crossing number is thus the number of crossings in the simplest picture of a knot. A diagram of a knot <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> with exactly <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c(K)</annotation></semantics></math> crossings is called a <em>minimal</em> diagram.</p></del><ins class='diffmod'><h2 id='definition'>Definition</h2></ins> <del class='diffmod'><h5 id='examples'>Examples</h5></del><ins class='diffmod'><p>Let <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a>.</p></ins> <del class='diffmod'><ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>unknot</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>c(unknot) = 0</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>trefoil</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>c(trefoil) = 3</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>figure</mi><mo>−</mo><mn>8</mn><mo stretchy='false'>)</mo><mo>=</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>c(figure-8) = 4</annotation></semantics></math>.</p> </li> <li> <p>if a diagram has <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math> crossings it represents the unknot, so there are no knots with <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>c(K) = 1</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_ed69b1768f4e5772a67714622539ad96e9a408f8_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>.</p> </li> </ul></del><ins class='diffmod'><div class='un_defn'> <h6 id='definition_2'>Definition</h6> <p>The <em>crossing number</em>, <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c(K)</annotation></semantics></math>, of <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> is the minimum number of crossings in a <a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagram</a> in the <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy class</a> of <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>.</p> </div></ins> <ins class='diffins'><p>The crossing number is thus the number of crossings in the simplest picture of a knot. A diagram of a knot <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> with exactly <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c(K)</annotation></semantics></math> crossings is called a <em>minimal</em> diagram.</p></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='examples'>Examples</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/unknot'>unknot</a>: <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>unknot</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>c(unknot) = 0</annotation></semantics></math>;</p> </li> <li> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/trefoil+knot'>trefoil knot</a>: <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>trefoil</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>c(trefoil) = 3</annotation></semantics></math>;</p> </li> <li> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/figure+eight+knot'>figure eight knot</a>: <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>figure</mi><mo>−</mo><mn>8</mn><mo stretchy='false'>)</mo><mo>=</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>c(figure-8) = 4</annotation></semantics></math>.</p> </li> <li> <p>if a <a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagram</a> has <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math> crossings it represents the <a class='existingWikiWord' href='/nlab/show/diff/unknot'>unknot</a>, so there are no non-trivial knots with <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>c(K) = 1</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_d0966a28a608c4e3f51b21c57e016f75ba95092a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>.</p> </li> </ul></ins><ins class='diffins'> </ins><p>The crossing number is related to the <a class='existingWikiWord' href='/nlab/show/diff/unknotting+number'>unknotting number</a>, but in quite a subtle way.</p> <del class='diffmod'><p>In the books by Burde and Zeischang (1985) and Kauffman (1987), the tables of knots are arranged according to crossing number. (Choices have been made of one <span class='newWikiWord'>mirror image<a href='/nlab/new/mirror+image'>?</a></span> or the other.) Given some arbitrary diagram, the crossing number of the knot that it respresents may be hard to determine.</p></del><ins class='diffmod'><h2 id='related_concepts'>Related concepts</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/writhe'>writhe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/linking+number'>linking number</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/framed+link'>framing number</a></p> </li> </ul></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='references'>References</h2></ins><ins class='diffins'> </ins><ins class='diffins'><p>In the books by Burde and Zeischang (1985) and Kauffman (1987), the tables of knots are arranged according to crossing number. (Choices have been made of one mirror image or the other.) Given some arbitrary diagram, the crossing number of the knot that it respresents may be hard to determine.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 18:32:31. 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