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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 #14 to #15: <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='#basics'>Basics</a></li><li><a href='#examples'>Examples</a><ul><li><a href='#trivial_links'>Trivial links</a></li><li><a href='#the_hopf_link'>The Hopf Link</a></li><li><a href='#the_borromean_link'>The Borromean Link</a></li><li><a href='#the_whitehead_link'>The Whitehead Link</a></li><li><a href='#brunnian_links'>Brunnian links</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <strong>link</strong> is a generalisation of a <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a> where one is allowed more than one component. Many <a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>knot invariants</a> extend to <a class='existingWikiWord' href='/nlab/show/diff/link+invariant'>link invariant</a>s and for many such invariants, one needs to know this extension to compute it even for a knot. Thus the study of links and knots is inextricably intertwined.</p> <ins class='diffins'><p>A link is equivalently a <a class='existingWikiWord' href='/nlab/show/diff/tangle'>tangle</a> without <a class='existingWikiWord' href='/nlab/show/diff/manifold+with+boundary'>boundary</a>.</p></ins><ins class='diffins'> </ins><h2 id='basics'>Basics</h2> <div class='num_defn' id='link'> <h6 id='definition'>Definition</h6> <p>A <strong>link</strong> is an <a class='existingWikiWord' href='/nlab/show/diff/embedding'>embedding</a> of a <a class='existingWikiWord' href='/nlab/show/diff/finite+number'>finite number</a> of copies of the <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>. The embedding is usually taken in <math class='maruku-mathml' display='inline' id='mathml_dc3d6af5036eeba9ae2cef57f5482bb48f2a5083_1' 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>, or its <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a>, <math class='maruku-mathml' display='inline' id='mathml_dc3d6af5036eeba9ae2cef57f5482bb48f2a5083_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>S^3</annotation></semantics></math>.</p> </div> <p>It is possible to generalise this to more varied sources and targets.</p> <p>Links can be studied in a number of ways depending on the notion of equivalence that is used. Coming from knot theory, one considers equivalence up to <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a>; that is, two links are equivalent if there is a homotopy between them which is an embedding for all times. A weaker notion was consider by <a href='#jmLinkGroups'>Milnor</a> wherein the components of the link are allowed to pass through themselves, but not through other components. That is, when restricted to each component it must be an immersion for all times, and the images of the components must always be disjoint.</p> <h2 id='examples'>Examples</h2> <h3 id='trivial_links'>Trivial links</h3> <p>Any <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a> is a link, and any <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of <a class='existingWikiWord' href='/nlab/show/diff/unknot'>unknot</a>s (called an <strong>unlink</strong>) is a link. We may call these ‘trivial’ (hopefully this name isn't standard for something different), in the sense of what you would know about before you study links.</p> <h3 id='the_hopf_link'>The Hopf Link</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Hopf+link'>Hopf link</a> is the simplest non-trivial (in the sense above) link, consisting of two components linked once.</p> <svg height='123.81097pt' viewBox='-5.0 -61.90549 180.71646 123.81097 ' width='180.71646pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'> <g transform='translate(0, 61.90549 ) scale(1,-1) translate(0,61.90549 )'> <g> <g stroke='rgb(0.0%,0.0%,0.0%)'> <g fill='rgb(0.0%,0.0%,0.0%)'> <g stroke-width='0.4pt'> <g> </g> <g> <g> <g stroke-width='4.0pt'> <g stroke='rgb(100.0%,100.0%,100.0%)'> <g fill='rgb(100.0%,100.0%,100.0%)'> <g> <g stroke-width='10.0pt'> <path d=' M 0.0 0.0 C 0.0 31.42844 25.47705 56.90549 56.90549 56.90549 C 88.33392 56.90549 113.81097 31.42844 113.81097 0.0 ' style='fill: none;' /> <g> <g stroke-width='2.0pt'> <g stroke='rgb(0.0%,50.0%,0.0%)'> <path d=' M 0.0 0.0 C 0.0 31.42844 25.47705 56.90549 56.90549 56.90549 C 88.33392 56.90549 113.81097 31.42844 113.81097 0.0 ' style='fill: none;' /> </g> </g> </g> </g> </g> </g> </g> </g> </g> <g> <g stroke-width='4.0pt'> <g stroke='rgb(100.0%,100.0%,100.0%)'> <g fill='rgb(100.0%,100.0%,100.0%)'> <g> <g stroke-width='10.0pt'> <path d=' M 113.81097 0.0 M 170.71646 0.0 C 170.71646 31.42844 145.23941 56.90549 113.81097 56.90549 C 82.38254 56.90549 56.90549 31.42844 56.90549 0.0 C 56.90549 -31.42844 82.38254 -56.90549 113.81097 -56.90549 C 145.23941 -56.90549 170.71646 -31.42844 170.71646 0.0 Z M 113.81097 0.0 ' style='fill: none;' /> <g> <g stroke-width='2.0pt'> <g stroke='rgb(100.0%,0.0%,0.0%)'> <path d=' M 113.81097 0.0 M 170.71646 0.0 C 170.71646 31.42844 145.23941 56.90549 113.81097 56.90549 C 82.38254 56.90549 56.90549 31.42844 56.90549 0.0 C 56.90549 -31.42844 82.38254 -56.90549 113.81097 -56.90549 C 145.23941 -56.90549 170.71646 -31.42844 170.71646 0.0 Z M 113.81097 0.0 ' style='fill: none;' /> </g> </g> </g> </g> </g> </g> </g> </g> </g> <g> <g stroke-width='4.0pt'> <g stroke='rgb(100.0%,100.0%,100.0%)'> <g fill='rgb(100.0%,100.0%,100.0%)'> <g> <g stroke-width='10.0pt'> <path d=' M 0.0 0.0 C 0.0 -31.42844 25.47705 -56.90549 56.90549 -56.90549 C 88.33392 -56.90549 113.81097 -31.42844 113.81097 0.0 ' style='fill: none;' /> <g> <g stroke-width='2.0pt'> <g stroke='rgb(0.0%,50.0%,0.0%)'> <path d=' M 0.0 0.0 C 0.0 -31.42844 25.47705 -56.90549 56.90549 -56.90549 C 88.33392 -56.90549 113.81097 -31.42844 113.81097 0.0 ' style='fill: none;' /> </g> </g> </g> </g> </g> </g> </g> </g> </g> </g> <g> </g> </g> </g> </g> </g> </g> </svg> <h3 id='the_borromean_link'>The Borromean Link</h3> <p>It is possible to link together <math class='maruku-mathml' display='inline' id='mathml_dc3d6af5036eeba9ae2cef57f5482bb48f2a5083_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> circles in such a way that removing any one makes the others fall apart. For <math class='maruku-mathml' display='inline' id='mathml_dc3d6af5036eeba9ae2cef57f5482bb48f2a5083_4' 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>, we have the Hopf link above; for <math class='maruku-mathml' display='inline' id='mathml_dc3d6af5036eeba9ae2cef57f5482bb48f2a5083_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>n = 3</annotation></semantics></math>, we have the <a class='existingWikiWord' href='/nlab/show/diff/Borromean+link'>Borromean link</a>, or Borromean Rings.</p> <svg height='173.0927pt' viewBox='-61.90549 -61.90549 180.71646 173.0927 ' width='180.71646pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'> <g transform='translate(0, 111.18721 ) scale(1,-1) translate(0,61.90549 )'> <g> <g stroke='rgb(0.0%,0.0%,0.0%)'> 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</g> </g> </g> </g> </g> </svg> <h3 id='the_whitehead_link'>The Whitehead Link</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+link'>Whitehead link</a> is an example of a link that shows the difference between the two notions of equivalence. If the links are only allowed to move by isotopies, then the two components are linked. However, if a link is allowed to pass through itself, then they can be unlinked.</p> <svg height='189.9509pt' viewBox='-156.29964 -94.97545 312.59927 189.9509 ' width='312.59927pt' xmlns:xlink='http://www.w3.org/1999/xlink' xmlns='http://www.w3.org/2000/svg'> <g transform='translate(0, 94.97545 ) scale(1,-1) translate(0,94.97545 )'> <g> <g stroke='rgb(0.0%,0.0%,0.0%)'> <g fill='rgb(0.0%,0.0%,0.0%)'> <g stroke-width='0.4pt'> <g> </g> <g> <g> <g stroke-width='4.0pt'> <g stroke='rgb(100.0%,100.0%,100.0%)'> <g fill='rgb(100.0%,100.0%,100.0%)'> <g> <g stroke-width='10.0pt'> <path d=' M 0.0 0.0 M 89.97545 0.0 C 89.97545 49.6927 49.6927 89.97545 0.0 89.97545 C -49.6927 89.97545 -89.97545 49.6927 -89.97545 0.0 C -89.97545 -49.6927 -49.6927 -89.97545 0.0 -89.97545 C 49.6927 -89.97545 89.97545 -49.6927 89.97545 0.0 Z M 0.0 0.0 ' style='fill: none;' /> <g> <g stroke-width='2.0pt'> <g stroke='rgb(100.0%,0.0%,0.0%)'> <path d=' M 0.0 0.0 M 89.97545 0.0 C 89.97545 49.6927 49.6927 89.97545 0.0 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</g> <g> </g> </g> </g> </g> </g> </g> </svg> <h3 id='brunnian_links'>Brunnian links</h3> <p>A <strong>Brunnian link</strong> is a link which is not an unlink but which has the property that the removal of any of its components results in an unlink. Technically, this includes the Hopf link and any knot (thanks to <a href='http://mathoverflow.net/questions/40724/is-the-hopf-link-a-brunnian-link'>this MO question</a> for settling that issue). The Borromean rings above are an example of a Brunnian link with three components.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+link'>hyperbolic link</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Milnor'>John Milnor</a>, (1954). Link groups. <em>Ann. of Math. (2)</em>, <em>59</em>, 177–195. <a href='http://www.ams.org/mathscinet-getitem?mr=71020' id='jmLinkGroups'>MR</a></p> </li> <li id='Birman75'> <p><a class='existingWikiWord' href='/nlab/show/diff/Joan+S.+Birman'>Joan S. Birman</a>, <em>Braids, links, and mapping class groups</em>, Princeton Univ Press (1975) [[ISBN:9780691081496](https://press.princeton.edu/books/paperback/9780691081496/braids-links-and-mapping-class-groups-am-82-volume-82), <a href='https://api.pageplace.de/preview/DT0400.9781400881420_A26691398/preview-9781400881420_A26691398.pdf'>preview pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dale+Rolfsen'>Dale Rolfsen</a>, <em>Knots and links</em>, AMS Chelsea <strong>346</strong> (2003) [[ams:chel-346-h](https://bookstore.ams.org/chel-346-h), <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/rolfsen.pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Manifold+Atlas'>Manifold Atlas</a>: <em><a href='http://www.map.mpim-bonn.mpg.de/High_codimension_links'>High codimension links</a></em></p> </li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 31, 2024 at 18:22:05. See the <a href="/nlab/history/link" 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/link" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/17344/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/link/14" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/link" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/link" accesskey="S" class="navlink" id="history" rel="nofollow">History (14 revisions)</a> <a href="/nlab/show/link/cite" style="color: black">Cite</a> <a href="/nlab/print/link" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/link" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>