Algebraic & Geometric Topology Volume 6, issue 5 (2006)

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <meta name="citation_title" content= "Knots with unknotting number 1 and essential Conway spheres" /> <meta name="citation_author" content="Gordon, Cameron McA" /> <meta name="citation_author_institution" content= "Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA" /> <meta name="citation_author" content="Luecke, John" /> <meta name="citation_author_institution" content= "Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA" /> <meta name="citation_publisher" content= "Mathematical Sciences Publishers" /> <meta name="citation_journal_title" content= "Algebraic & Geometric Topology" /> <meta name="citation_volume" content="6" /> <meta name="citation_issue" content="5" /> <meta name="citation_firstpage" content="2051" /> <meta name="citation_lastpage" content="2116" /> <meta name="citation_publication_date" content="2006/11/19" /> <meta name="citation_pdf_url" content= "http://msp.org/agt/2006/6-5/agt-v6-n5-p02-s.pdf" /> <meta name="citation_fulltext_world_readable" content="" /> <meta name="citation_doi" content="10.2140/agt.2006.6.2051" /> <meta name="citation_issn" content="1472-2739" /> <title>Algebraic & Geometric Topology Volume 6, issue 5 (2006)</title> <link href="/agt/etc/journal.css" type="text/css" rel="stylesheet" /> <link href="/agt/etc/abstract.css" type="text/css" rel="stylesheet" /> <link rel="shortcut icon" href="/agt/etc/favicon.ico" /> <link href="/agt/rss/rss.xml" rel="alternate" type="application/rss+xml" title="Algebraic & Geometric Topology" /> <script type="text/javascript" src="/agt/etc/cover.js"> </script> <script type="text/javascript" src="/agt/etc/journal.js"> </script> <script 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class="content-column" id="content-area"> <table cellspacing="0" cellpadding="0"> <tr> <td class="title-area"> <a class="title" href= "/agt/2006/6-5/agt-v6-n5-p02-s.pdf">Knots with unknotting number 1 and essential Conway spheres</a> <h3>Cameron McA Gordon and John Luecke</h3> </td> </tr> <tr> <td> <div class="page-numbers-heading"> Algebraic & Geometric Topology 6 (2006) 2051–2116 </div> </td> </tr> <tr> <td> <div class="paper-doi"> DOI: <a href= "http://dx.doi.org/10.2140/agt.2006.6.2051">10.2140/agt.2006.6.2051</a> </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <p class="noindent">arXiv: <a href= "http://arxiv.org/abs/math.GT/0601265">math.GT/0601265</a></p> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Abstract</h5> </td> </tr> <tr> <td class="article-area"> <p class="noindent">For a knot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="bold"><mi>T</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math> be the characteristic toric sub-orbifold of the orbifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo class="MathClass-punc">,</mo><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math> as defined by Bonahon–Siebenmann. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math> has unknotting number one, we show that an unknotting arc for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math> can always be found which is disjoint from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="bold"><mi>T</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>, unless either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math> is an EM–knot (of Eudave-Muñoz) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo class="MathClass-punc">,</mo><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math> contains an EM–tangle after cutting along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle mathvariant="bold"><mi>T</mi></mstyle><mrow><mo class="MathClass-open">(</mo><mrow><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>. As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth–Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. </p><p class="indent"> As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation. </p> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Keywords</h5> <div class="keywords"> unknotting number 1, Conway spheres, algebraic knots, mutation </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Mathematical Subject Classification 2000</h5> <div class="keywords"> Primary: 57N10 </div> <div class="keywords"> Secondary: 57M25 </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5><a href="b02.xhtml">References</a></h5> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5><a href="f02.xhtml">Forward citations</a></h5> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Publication</h5> <div class="keywords"> Received: 9 January 2006 </div> <div class="keywords"> Accepted: 29 September 2006 </div> <div class="keywords"> Published: 19 November 2006 </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Authors</h5> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="author-contact"> <tr> <td class="author-name" colspan="2"><a href="#email" class= "author-name" onclick= "javascript: if(confirm('Would you like to see this person\'s email?')){ var a = Array(1038,26168,34381,2399,31800,42906,46536,62053,57134,26307,44199); var p = 65539; var b = Array(2 * a.length); for(var i = 0; i < a.length; i++){ var t = 1; for(var j = 0; j <65536; j++){ if (t == a[i]){ b[2*i] = String.fromCharCode(j & 255); b[2*i + 1] = String.fromCharCode((j >>> 8) & 255); j = 65536; }; t = (2 * t) % p ; }; } alert(b.join('')); } ;return false;"> Cameron McA Gordon</a></td> </tr> <tr> <td class="author-address">Department of Mathematics<br /> The University of Texas at Austin<br /> 1 University Station C1200<br /> Austin, TX 78712-0257<br /> USA</td> </tr> <tr> <td></td> </tr> <tr> <td class="author-name" colspan="2"><a href="#email" class= "author-name" onclick= "javascript: if(confirm('Would you like to see this person\'s email?')){ var a = Array(42236,52601,61549,2399,31800,42906,46536,62053,57134,26307,44199); var p = 65539; var b = Array(2 * a.length); for(var i = 0; i < a.length; i++){ var t = 1; for(var j = 0; j <65536; j++){ if (t == a[i]){ b[2*i] = String.fromCharCode(j & 255); b[2*i + 1] = String.fromCharCode((j >>> 8) & 255); j = 65536; }; t = (2 * t) % p ; }; } alert(b.join('')); } ;return false;"> John Luecke</a></td> </tr> <tr> <td class="author-address">Department of Mathematics<br /> The University of Texas at Austin<br /> 1 University Station C1200<br /> Austin, TX 78712-0257<br /> USA</td> </tr> <tr> <td></td> </tr> </table> </td> </tr> <tr> <td> <table cellspacing="0" cellpadding="0" class="pause"> <tr> <td class="action-end"><img src="/agt/etc/z.gif" alt="" /></td> </tr> </table> </td> <td id="content-area-end"></td> </tr> </table> <table cellspacing="0" cellpadding="0" id="footer-area"> <tr> <td class="box-footer" onclick= "javascript:window.location='http://msp.org/';"> <img src="/mod/images/msp-logo.png" alt= "Mathematical Sciences Publishers" /> <p class="copyright">© Copyright 2006 Mathematical Sciences Publishers. 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Algebraic & Geometric Topology Volume 6, issue 5 (2006)