Knot complement

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the knot complement of a <a href="/facts/Tame_knot/t996suvy">tame knot</a> K is the space where the knot is not. If a knot is embedded in the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>). Let N be a <a href="/facts/Tubular_neighborhood/4Ftas0ry">tubular neighborhood</a> of K; so N is a <a href="/facts/Solid_torus/WWEAGbU7">solid torus</a>. The knot complement is then the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> of N,

X
          
            K
          
        
        =
        M
        −
        
          
            interior
          
        
        (
        N
        )
        .
      
    
    {\displaystyle X_{K}=M-{\mbox{interior}}(N).}

The knot complement XK is a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/3-manifold/48yr0Tzx">3-manifold</a>; the boundary of XK and the boundary of the neighborhood N are homeomorphic to a two-<a href="/facts/Torus/J9SkaV1W">torus</a>. Sometimes the ambient manifold M is understood to be the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>. Context is needed to determine the usage. There are analogous definitions for the <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a> complement.
Many <a href="/facts/Knot_invariant/BEvEEpHQ">knot invariants</a>, such as the <a href="/facts/Knot_group/bremjfld">knot group</a>, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the <a href="/facts/Gordon%E2%80%93Luecke_theorem/X6N4w9KE">Gordon–Luecke theorem</a> states that a knot is determined by its complement. That is, if K and K′ are two knots with <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> complements then there is a homeomorphism of the three-sphere taking one knot to the other.
Knot complements are <a href="/facts/Haken_manifolds/ebWXqIvC">Haken manifolds</a>. More generally complements of <a href="/facts/Link_(knot_theory)/hoFDYmVj">links</a> are Haken manifolds.

Knot complement open-in-new

Knot complement