CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Milnor mu-bar invariant (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Milnor mu-bar invariant (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2510/#Item_5" title="Discuss this page in its dedicated thread on the nForum" 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 #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='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</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='#link_group'>Link Group</a></li><li><a href='#invariants'><math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math>-Invariants</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p><del class='diffmod'>In </del><ins class='diffmod'><a href='#Milnor64'>Milnor 1964</a></ins><del class='diffmod'><em><a href='#jmLinkGroups'>Link Groups</a></em></del><ins class='diffmod'> introduced the notion of the </ins><del class='diffmod'> , </del><ins class='diffmod'><em>link group</em></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/John+Milnor'>John Milnor</a></del><ins class='diffmod'> as a way to study </ins><del class='diffdel'> introduced the notion of the </del><del class='diffdel'><span class='newWikiWord'>Link Group<a href='/nlab/new/Link+Group'>?</a></span></del><del class='diffdel'> as a way to study </del><a class='existingWikiWord' href='/nlab/show/diff/link'>links</a>. The notion of equivalence of links that Milnor used is slightly different to that obtained by extending the usual notion of equivalence of <a class='existingWikiWord' href='/nlab/show/diff/knot'>knots</a>. In Milnor’s paper, the crucial aspect of links was the interactions between distinct components. Thus for Milnor, a link in a manifold <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is a map <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mi>n</mi></msub><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>\coprod_n S^1 \to M</annotation></semantics></math> such that the components have disjoint images. Similarly, two links are <strong>homotopic</strong> if there is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a> between the maps which is a link at every time. Thus links can be deformed in such a manner that individual components can pass through themselves, but not through other components. Also link components can have self-intersections or the map on a component can be a constant map. Milnor uses the term <strong>proper link</strong> to refer to a link in which the map is a homeomorphism onto its image.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+link'>Whitehead link</a> is a simple example of a link that is not trivial under ambient <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a> but is trivial under Milnor’s notion of homotopy.</p> <p>The <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math><span><del class='diffmod'> -invariants</del><ins class='diffmod'> -</ins><del class='diffdel'> come</del><del class='diffdel'> from</del><del class='diffdel'> explicit</del><del class='diffdel'> descriptions</del><del class='diffdel'> of</del><del class='diffdel'> the</del><del class='diffdel'> link</del><del class='diffdel'> groups</del><del class='diffdel'> of</del><del class='diffdel'> particular</del><del class='diffdel'> links.</del><del class='diffdel'> Specifically,</del><del class='diffdel'> Milnor</del><del class='diffdel'> calls</del><del class='diffdel'> a</del><del class='diffdel'> link</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/invariant'>invariants</a></ins><ins class='diffins'> come from explicit descriptions of the link groups of particular links. Specifically, Milnor calls a link </ins><em>almost trivial</em> if every proper sublink is trivial (see <a class='existingWikiWord' href='/nlab/show/diff/link'>Brunnian link</a>). Such a link corresponds to an element in a particular link group which can be completely described by certain numbers.</p> <h2 id='link_group'>Link Group</h2> <p>Let us begin by describing the link group. Milnor’s alternative description is as follows. Consider the complement of a link <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> in an open <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-manifold <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. We choose a <a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>basepoint</a> in this complement and so have the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>. We define a relation on this group as follows: two loops <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math> are equivalent if the link <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>∪</mo><msup><mi>α</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mi>β</mi></mrow><annotation encoding='application/x-tex'>L \cup \alpha^{-1} \beta</annotation></semantics></math> is homotopic in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> to one of the form <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mo>′</mo><mo>∪</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>L' \cup 1</annotation></semantics></math> (where <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> is the constant loop at the basepoint). The <strong>link group</strong> is the group of equivalence classes of such loops.</p> <p>A more practical description is the following.</p> <div class='num_defn' id='linkgroup'> <h6 id='definition'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be a link in an open <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-manifold <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G(L)</annotation></semantics></math> be the fundamental group of the complement of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>L^i</annotation></semantics></math> denote the sublink obtained by deleting the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th component of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A_i(L)</annotation></semantics></math> be the kernel of the natural inclusion <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>G</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G(L) \to G(L^i)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[A_i]</annotation></semantics></math> its commutator subgroup. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>[</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo stretchy='false'>[</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo><mi>⋯</mi><mo stretchy='false'>[</mo><msub><mi>A</mi> <mi>n</mi></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>E(L) = [A_1] [A_2] \cdots [A_n]</annotation></semantics></math>. This is a normal subgroup of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G(L)</annotation></semantics></math>. The quotient, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>E</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L) \coloneqq G(L)/E(L)</annotation></semantics></math> is the <strong>link group</strong> of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>.</p> </div> <p>Milnor’s first theorem on this group was to show that this group is an invariant of the homotopy class of the link, at least for proper links.</p> <div class='num_theorem' id='htyinv'> <h6 id='theorem_milnor_theorem_1'>Theorem (<a href='#jmLinkGroups'>Milnor, Theorem 1</a>)</h6> <p>If two proper links are homotopic, then their link groups are isomorphic.</p> </div> <p>To study this group for a particular link, we need to find some particular elements in it. These are the <strong>meridians</strong> and the <strong>parallels</strong>. Basically, a meridian goes around one component of the link once, in a specific direction, whilst a parallel goes along it. Technically, the parallels of a link are not <em>elements</em> of its link group, but cosets.</p> <p>Choose a component <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math> of the link <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. Choose orientations of the ambient manifold, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, and of the circle. To define the meridian and parallel of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math> we need to choose a path from the basepoint, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math>, to a point on the image of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math> which does not intersect the image of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> at any other time. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> be such a path, so then <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p(1)</annotation></semantics></math> is a point on the image of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math>.</p> <div class='num_defn' id='meridian'> <h6 id='definition_2'>Definition</h6> <p>The <strong><math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th meridian</strong> of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is the element <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub><mo>∈</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha_i \in \mathcal{G}(L)</annotation></semantics></math> defined as follows. Choose a small neighbourhood <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p(1)</annotation></semantics></math>. Define a path by going along <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> until we are inside <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math>, then go around a closed loop in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> which has linking number <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>+1</annotation></semantics></math> with the part of the image of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math> inside <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math>. Then return to <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>.</p> </div> <div class='num_defn' id='parallel'> <p>The <strong><math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th parallel</strong> of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is the coset <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>β</mi> <mi>i</mi></msub><msub><mi>𝒜</mi> <mi>i</mi></msub><mo>∈</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\beta_i \mathcal{A}_i \in \mathcal{G}(L^i)</annotation></semantics></math> defined as follows. The subgroup <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{A}_i</annotation></semantics></math> is the kernel of the homomorphism <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L) \to \mathcal{G}(L^i)</annotation></semantics></math>. Go along <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> to its end. Then go around the image of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>L_i</annotation></semantics></math> according to the orientation of the circle. Finally return to <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>. The preimage of this element defines a coset in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L)</annotation></semantics></math> which we write as <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>β</mi> <mi>i</mi></msub><msub><mi>𝒜</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\beta_i \mathcal{A}_i</annotation></semantics></math>.</p> </div> <p>The basic method of studying a link via link groups is to consider a link as an element of the link group of the link obtained by removing one of its components. To show that this is a reasonable thing to do, Milnor proved the following theorem.</p> <div class='num_theorem' id='conjelts'> <h6 id='theorem_milnor_theorem_3'>Theorem (<a href='#jmLinkGroups'>Milnor, Theorem 3</a>)</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be a proper link with <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> components. Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>f'</annotation></semantics></math> be closed loops in the complement of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. If they represent conjugate elements of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L)</annotation></semantics></math> then the links <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>,</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(L,f)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>L</mi><mo>,</mo><mi>f</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(L,f')</annotation></semantics></math> are homotopic.</p> </div> <h2 id='invariants'><math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math>-Invariants</h2> <p>For <a class='existingWikiWord' href='/nlab/show/diff/link'>Brunnian links</a>, which Milnor calls <strong>almost trivial</strong> links, the classification question reduces to looking at elements of the link group of trivial links. It is important to note that the ambient space here is Euclidean space, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_72' 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>.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be an <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-component Brunnian link. Then we consider the element <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><msub><mo>′</mo> <mi>n</mi></msub><mo>∈</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\beta'_n \in \mathcal{G}(L^n)</annotation></semantics></math> corresponding to the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th parallel. Upon removing a further component, say the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n-1</annotation></semantics></math>st, this element becomes trivial since we are then looking at <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>L^{n-1}</annotation></semantics></math> which is trivial. Thue <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><msub><mo>′</mo> <mi>n</mi></msub><mo>∈</mo><msub><mi>𝒜</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\beta'_n \in \mathcal{A}_{n-1}(L^n)</annotation></semantics></math>, the kernel of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo>→</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L^n) \to \mathcal{G}(L^{n-1,n})</annotation></semantics></math> (here <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>L^{n-1,n}</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> with both the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n-1</annotation></semantics></math>st and <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th components removed). Now <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A}_{n-1}(L^n)</annotation></semantics></math> is the smallest normal subgroup containing the meridian <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\alpha_{n-1}</annotation></semantics></math> (since removing the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n-1</annotation></semantics></math>st component is the same thing as allowing the meridian <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\alpha_{n-1}</annotation></semantics></math> to collapse) and so every element of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{A}_{n-1}</annotation></semantics></math> can be written as a word in alphabet of powers and conjugates of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\alpha_{n-1}</annotation></semantics></math>. Milnor uses the notation</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mi>σ</mi></msubsup><mo>,</mo><mspace width='1em' /><mi>σ</mi><mo>∈</mo><mi>ℛ</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \alpha_{n-1}^\sigma, \quad \sigma \in \mathcal{R}(L^{n-1,n}) </annotation></semantics></math></div> <p>to write this. The exponent <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> itself decomposes as</p> <div class='maruku-equation' id='eq:muinvs'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>=</mo><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mi>n</mi><mo stretchy='false'>)</mo><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> \sigma = \sum \mu(i_1 \cdots i_{n-2}, n-1 n) k_{i_1} \cdots k_{i_{n-2}} </annotation></semantics></math></div> <p>where the summation is over all permutations <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>i_1 \cdots i_{n-2}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>, …, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n-2</annotation></semantics></math>.</p> <div class='num_theorem' id='muinvs'> <h6 id='theorem_milnor_section_5'>Theorem (<a href='#jmLinkGroups'>Milnor, Section 5</a>)</h6> <p>The integers <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mu(i_1 \cdots i_{n-2}, n_1 n)</annotation></semantics></math> are homotopy invariants of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. The homotopy class of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is completely specified by these integers.</p> </div> <p>There was nothing special about the choice of components. A similar procedure works for any pair of components. The resulting integers obey the following rules:</p> <div class='maruku-equation' id='eq:eq2'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><msub><mi>i</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mi>n</mi></msub><msub><mi>i</mi> <mn>1</mn></msub><msub><mi>i</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub><mo>,</mo><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mi>ν</mi></msub><mi>r</mi><msub><mi>j</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>j</mi> <mrow><mi>n</mi><mo>−</mo><mi>ν</mi><mo>−</mo><mn>2</mn></mrow></msub><mi>s</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mi>n</mi><mo>−</mo><mi>ν</mi></mrow></msup><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mi>μ</mi><mo stretchy='false'>(</mo><msub><mi>h</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>h</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mi>r</mi><mi>s</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} \mu(i_1 i_2 \cdots i_{n-2}, i_{n-1} i_n) &= \mu(i_n i_1 i_2 \cdots i_{n-3}, i_{n-2} i_{n-2}) \\ \mu(i_1 \cdots i_{\nu} r j_1 \cdots j_{n-\nu-2} s) &= (-1)^{n-\nu} \sum \mu(h_1 \cdots h_{n-2} r s) \end{aligned} </annotation></semantics></math></div> <p>In the second identity, the summation is over all shuffle products of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>i</mi> <mi>ν</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(i_1 \cdots i_{\nu})</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>j</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>j</mi> <mrow><mi>n</mi><mo>−</mo><mi>ν</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(j_1 \cdots j_{n - \nu - 2})</annotation></semantics></math>.</p> <p>Let us expand on the definition of the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math>-invariants. We start with the exponential notation. The following holds for an arbitrary proper link, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>, embedded in an open <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math>-manifold <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>J \mathcal{G}(L)</annotation></semantics></math> be the integral group ring of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{G}(L)</annotation></semantics></math>. As mentioned above, any element of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A}_i(L)</annotation></semantics></math> is a product of powers of conjugates of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\alpha_i</annotation></semantics></math>. We can write such an element in the form <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>α</mi> <mi>i</mi> <mi>s</mi></msubsup></mrow><annotation encoding='application/x-tex'>\alpha_i^s</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi><mo>∈</mo><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>s \in J \mathcal{G}(L)</annotation></semantics></math> by interpreting:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msubsup><mi>α</mi> <mi>i</mi> <mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></msubsup></mtd> <mtd><mo>=</mo><msubsup><mi>α</mi> <mi>i</mi> <mi>x</mi></msubsup><msubsup><mi>α</mi> <mi>i</mi> <mi>y</mi></msubsup></mtd></mtr> <mtr><mtd><msubsup><mi>α</mi> <mi>i</mi> <mrow><mi>k</mi><mi>x</mi></mrow></msubsup></mtd> <mtd><mo>=</mo><mo stretchy='false'>(</mo><msubsup><mi>α</mi> <mi>i</mi> <mi>x</mi></msubsup><msup><mo stretchy='false'>)</mo> <mi>k</mi></msup></mtd></mtr> <mtr><mtd><msubsup><mi>α</mi> <mi>i</mi> <mi>β</mi></msubsup></mtd> <mtd><mo>=</mo><mi>β</mi><mi>α</mi><msup><mi>β</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} \alpha_i^{x + y} &= \alpha_i^x \alpha_i^y \\ \alpha_i^{k x} &= (\alpha_i^x)^k \\ \alpha_i^\beta &= \beta \alpha \beta^{-1} \end{aligned} </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x, y \in J\mathcal{G}(L)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{Z}</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mo>∈</mo><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\beta \in \mathcal{G}(L)</annotation></semantics></math>.</p> <p>We write the kernel of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>J \mathcal{G}(L) \to J \mathcal{G}(L^i)</annotation></semantics></math> as <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒦</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{K}_i(L)</annotation></semantics></math>. Using these, we define:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℛ</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><msub><mi>𝒦</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>L</mi><msup><mo stretchy='false'>)</mo> <mn>2</mn></msup><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>𝒦</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><msup><mo stretchy='false'>)</mo> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'> \mathcal{R}(L) \coloneqq J \mathcal{G}(L) / \mathcal{K}_1(L)^2 + \cdots + \mathcal{K}_n(L)^2 </annotation></semantics></math></div> <p>Now the notation <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>α</mi> <mi>i</mi> <mi>s</mi></msubsup></mrow><annotation encoding='application/x-tex'>\alpha_i^s</annotation></semantics></math> for an element of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A}_i(L)</annotation></semantics></math> does not provide an injective map from <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mi>𝒢</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>J\mathcal{G}(L)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A}_i(L)</annotation></semantics></math>. The kernel is the ideal <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒦</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo><mo>+</mo><mo stretchy='false'>(</mo><msub><mi>𝒦</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>L</mi><msup><mo stretchy='false'>)</mo> <mn>2</mn></msup><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>𝒦</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><msup><mo stretchy='false'>)</mo> <mn>2</mn></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{K}_i(L) + (\mathcal{K}_1(L)^2 + \cdots + \mathcal{K}_n(L)^2)</annotation></semantics></math> which is naturally isomorphic to <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℛ</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{R}(L^i)</annotation></semantics></math>.</p> <div class='num_theorem' id='trivlink'> <h6 id='theorem_milnor_theorem_5'>Theorem (<a href='#jmLinkGroups'>Milnor, Theorem 5</a>)</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be a link which is homotopic to one in with the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th component is constant. Then every element of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒜</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{A}_i(L)</annotation></semantics></math> can be expressed uniquely in the form <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>α</mi> <mi>i</mi> <mi>σ</mi></msubsup></mrow><annotation encoding='application/x-tex'>\alpha_i^\sigma</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>∈</mo><mi>ℛ</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>i</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\sigma \in \mathcal{R}(L^i)</annotation></semantics></math>.</p> </div> <p>Now let us suppose that <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> is trivial. Then <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>G(L)</annotation></semantics></math> is the free product of the fundamental group of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> with the infinite cyclic groups generated by the (elements representing the) meridians of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math>. Let these be <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>a_1</annotation></semantics></math>, …, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>a_n</annotation></semantics></math> and let <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k_i = a_i - 1</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mi>G</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>J G(L)</annotation></semantics></math>. Milnor defines a <strong>canonical word</strong> to be a product of the form <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub><msub><mi>k</mi> <mrow><msub><mi>j</mi> <mn>1</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mn>1</mn></msub><msub><mi>k</mi> <mrow><msub><mi>j</mi> <mn>2</mn></msub></mrow></msub><msub><mi>ϕ</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>k</mi> <mrow><msub><mi>j</mi> <mi>p</mi></msub></mrow></msub><msub><mi>ϕ</mi> <mi>p</mi></msub></mrow><annotation encoding='application/x-tex'>\phi_0 k_{j_1} \phi_1 k_{j_2} \phi_2 \cdots k_{j_p} \phi_p</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>p \ge 0</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\phi_i \in \pi_1(M)</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>≤</mo><msub><mi>j</mi> <mi>i</mi></msub><mo>≤</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>1 \le j_i \le n</annotation></semantics></math>. A <strong>canonical sentence</strong> is a sum or difference of any number of canonical words. It turns out (<a href='#jmLinkGroups'>Milnor, Theorem 7</a>) that each element of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℛ</mi><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{R}(L)</annotation></semantics></math> is represented by a unique canonical sentence.</p> <p>Now let us return to the case of the almost trivial link in Euclidean space. From above, we have the element <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><msub><mo>′</mo> <mi>n</mi></msub><mo>∈</mo><msub><mi>𝒜</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\beta'_n \in \mathcal{A}_i(L^n)</annotation></semantics></math> corresponding to the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th parallel. Removing any other component allows us to trivialise <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><msub><mo>′</mo> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\beta'_n</annotation></semantics></math> since removing, say, the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>th component leaves us with <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mi>i</mi></msup></mrow><annotation encoding='application/x-tex'>L^i</annotation></semantics></math> which is homotopic to the trivial link on <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n-1</annotation></semantics></math> components. Removing the <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> component corresponds to setting <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>a_i</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mi>G</mi><mo stretchy='false'>(</mo><msup><mi>L</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>J G(L^n)</annotation></semantics></math>, equivalently to setting <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>k_i = 0</annotation></semantics></math>. So upon setting <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>k_i = 0</annotation></semantics></math> we must have that <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><msub><mo>′</mo> <mi>n</mi></msub><mo>↦</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\beta'_n \mapsto 1</annotation></semantics></math> and thus (by uniqueness) <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>↦</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\sigma \mapsto 0</annotation></semantics></math>. Hence <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>k_i</annotation></semantics></math> divides <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>, and so every canonical word in <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> is of the form <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>k_{i_1} \cdots k_{i_{n-2}}</annotation></semantics></math> for some permutation of <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>, …, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n-2</annotation></semantics></math>. Sorting them out by permutation, we get the expression in <a class='maruku-eqref' href='#eq:muinvs'>(1)</a>.</p> <p>Now, how do we interpret or calculate these invariants? We need to work out what an expression of the form in <a class='maruku-eqref' href='#eq:muinvs'>(1)</a> is saying. Consider a canonical word, <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>k_{i_1} \cdots k_{i_{n-2}}</annotation></semantics></math>. The corresponding element is:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mrow><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mi>⋯</mi><msub><mi>k</mi> <mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msub></mrow></msubsup></mrow><annotation encoding='application/x-tex'> \alpha_{n-1}^{k_{i_1} \cdots k_{i_{n-2}}} </annotation></semantics></math></div> <p>Let us write <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>=</mo><msub><mi>α</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\alpha = \alpha_{n-1}</annotation></semantics></math>. Now <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>α</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding='application/x-tex'>\alpha^{k_1}</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>α</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msub><mi>a</mi> <mn>1</mn></msub><mi>α</mi><msubsup><mi>a</mi> <mn>1</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><msup><mi>α</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\alpha^{a_1 - 1} = a_1 \alpha a_1^{-1} \alpha^{-1}</annotation></semantics></math>. Thus this tells us to go around <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>L_1</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{n-1}</annotation></semantics></math>, back around <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>L_1</annotation></semantics></math>, and finally back around <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{n-1}</annotation></semantics></math>. Each time we introduce a new power, we do the same except that we replace the loop around <math class='maruku-mathml' display='inline' id='mathml_65ce73d3a13dab44a7b39543ba6e05366cedbe8c_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>L_{n-1}</annotation></semantics></math> with the loop so far constructed.</p> <p>So the general method is as follows: choose two components of the link. Write one of them as a word in the meridians of the others. Then simplify this word using the other chosen link as the “base”: namely, write everything in terms of conjugates of that base. This will then separate out into the desired form and, hopefully, the link invariants can be read off.</p> <h2 id='references'>References</h2> <ul> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/John+Milnor'>John Milnor</a></del><ins class='diffmod'> </ins><del class='diffmod'> (1954). Link groups. </del><ins class='diffmod'><p><a class='existingWikiWord' href='/nlab/show/diff/John+Milnor'>John Milnor</a>: <em>Link groups</em>, Ann. of Math. <strong>2</strong> 59 (1954) 177-195 [[doi:10.2307/1969685](https://doi.org/10.2307/1969685), <a href='https://www.jstor.org/stable/1969685'>jstor:1969685</a>, <a href='http://www.ams.org/mathscinet-getitem?mr=71020'>MR</a>]</p></ins><del class='diffmod'><em>Ann. of Math. (2)</em></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><em>59</em></del><del class='diffdel'>, 177–195. </del><del class='diffdel'><a href='http://www.ams.org/mathscinet-getitem?mr=71020' id='jmLinkGroups'>MR</a></del></li><ins class='diffins'> </ins><ins class='diffins'><li> <p>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Link_group'>Link group</a></em></p> </li></ins> </ul><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> </div> <div class="revisedby"> <p> Last revised on July 12, 2024 at 08:43:11. See the <a href="/nlab/history/Milnor+mu-bar+invariant" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Milnor+mu-bar+invariant" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2510/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/Milnor+mu-bar+invariant/3" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/Milnor+mu-bar+invariant" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/Milnor+mu-bar+invariant" accesskey="S" class="navlink" id="history" rel="nofollow">History (3 revisions)</a> <a href="/nlab/show/Milnor+mu-bar+invariant/cite" style="color: black">Cite</a> <a href="/nlab/print/Milnor+mu-bar+invariant" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Milnor+mu-bar+invariant" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>