# Signature of a knot

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.

Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. The Seifert form of S is the pairing ${\displaystyle \phi :H_{1}(S)\times H_{1}(S)\to \mathbb {Z} }$ given by taking the linking number ${\displaystyle lk(a^{+},b^{-})}$ where ${\displaystyle a,b\in H_{1}(S)}$ and ${\displaystyle a^{+},b^{-}}$ indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S.

Given a basis ${\displaystyle b_{1},...,b_{2g}}$ for ${\displaystyle H_{1}(S)}$ (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, ${\displaystyle V_{ij}=\phi (b_{i},b_{j})}$. The signature of the matrix ${\displaystyle V+V^{t}}$, thought of as a symmetric bilinear form, is the signature of the knot K.

Slice knots are known to have zero signature.

## The Alexander module formulation

Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let ${\displaystyle X}$  be the universal abelian cover of the knot complement. Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: ${\displaystyle H_{1}(X;\mathbb {Q} )}$ . Given a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ -module ${\displaystyle V}$ , let ${\displaystyle {\overline {V}}}$  denote the ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ -module whose underlying ${\displaystyle \mathbb {Q} }$ -module is ${\displaystyle V}$  but where ${\displaystyle \mathbb {Z} }$  acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for ${\displaystyle X}$  gives a canonical isomorphism ${\displaystyle H_{1}(X;\mathbb {Q} )\simeq {\overline {H^{2}(X;\mathbb {Q} )}}}$  where ${\displaystyle H^{2}(X;\mathbb {Q} )}$  denotes the 2nd cohomology group of ${\displaystyle X}$  with compact supports and coefficients in ${\displaystyle \mathbb {Q} }$ . The universal coefficient theorem for ${\displaystyle H^{2}(X;\mathbb {Q} )}$  gives a canonical isomorphism with ${\displaystyle Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])}$  (because the Alexander module is ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ -torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ -modules ${\displaystyle Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])\simeq Hom_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),[\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ])}$ , where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$  denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ . This isomorphism can be thought of as a sesquilinear duality pairing ${\displaystyle H_{1}(X;\mathbb {Q} )\times H_{1}(X;\mathbb {Q} )\to [\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ]}$  where ${\displaystyle [\mathbb {Q} [\mathbb {Z} ]]}$  denotes the field of fractions of ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ . This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a ${\displaystyle \mathbb {Q} [\mathbb {Z} ]}$ -module is isomorphic to ${\displaystyle \mathbb {Q} [\mathbb {Z} ]/\Delta K}$ . Let ${\displaystyle tr:\mathbb {Q} [\mathbb {Z} ]/\Delta K\to \mathbb {Q} }$  be any linear function which is invariant under the involution ${\displaystyle t\longmapsto t^{-1}}$ , then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on ${\displaystyle H_{1}(X;\mathbb {Q} )}$  whose signature is an invariant of the knot.

All such signatures are concordance invariants, so all signatures of slice knots are zero. The sesquilinear duality pairing respects the prime-power decomposition of ${\displaystyle H_{1}(X;\mathbb {Q} )}$ —i.e.: the prime power decomposition gives an orthogonal decomposition of ${\displaystyle H_{1}(X;\mathbb {R} )}$ . Cherry Kearton has shown how to compute the Milnor signature invariants from this pairing, which are equivalent to the Tristram-Levine invariant.