# 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 $\phi :H_{1}(S)\times H_{1}(S)\to \mathbb {Z}$ given by taking the linking number $lk(a^{+},b^{-})$ where $a,b\in H_{1}(S)$ and $a^{+},b^{-}$ indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S.

Given a basis $b_{1},...,b_{2g}$ for $H_{1}(S)$ (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, $V_{ij}=\phi (b_{i},b_{j})$ . The signature of the matrix $V+V^{t}$ , 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 $X$  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: $H_{1}(X;\mathbb {Q} )$ . Given a $\mathbb {Q} [\mathbb {Z} ]$ -module $V$ , let ${\overline {V}}$  denote the $\mathbb {Q} [\mathbb {Z} ]$ -module whose underlying $\mathbb {Q}$ -module is $V$  but where $\mathbb {Z}$  acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for $X$  gives a canonical isomorphism $H_{1}(X;\mathbb {Q} )\simeq {\overline {H^{2}(X;\mathbb {Q} )}}$  where $H^{2}(X;\mathbb {Q} )$  denotes the 2nd cohomology group of $X$  with compact supports and coefficients in $\mathbb {Q}$ . The universal coefficient theorem for $H^{2}(X;\mathbb {Q} )$  gives a canonical isomorphism with $Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])$  (because the Alexander module is $\mathbb {Q} [\mathbb {Z} ]$ -torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of $\mathbb {Q} [\mathbb {Z} ]$ -modules $Ext_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])\simeq Hom_{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),[\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ])$ , where $[\mathbb {Q} [\mathbb {Z} ]]$  denotes the field of fractions of $\mathbb {Q} [\mathbb {Z} ]$ . This isomorphism can be thought of as a sesquilinear duality pairing $H_{1}(X;\mathbb {Q} )\times H_{1}(X;\mathbb {Q} )\to [\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ]$  where $[\mathbb {Q} [\mathbb {Z} ]]$  denotes the field of fractions of $\mathbb {Q} [\mathbb {Z} ]$ . This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a $\mathbb {Q} [\mathbb {Z} ]$ -module is isomorphic to $\mathbb {Q} [\mathbb {Z} ]/\Delta K$ . Let $tr:\mathbb {Q} [\mathbb {Z} ]/\Delta K\to \mathbb {Q}$  be any linear function which is invariant under the involution $t\longmapsto t^{-1}$ , then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on $H_{1}(X;\mathbb {Q} )$  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 $H_{1}(X;\mathbb {Q} )$ —i.e.: the prime power decomposition gives an orthogonal decomposition of $H_{1}(X;\mathbb {R} )$ . Cherry Kearton has shown how to compute the Milnor signature invariants from this pairing, which are equivalent to the Tristram-Levine invariant.