# 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 $\operatorname {lk} (a^{+},b^{-})$ 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 $\operatorname {Ext} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])$ (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 $\operatorname {Ext} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),\mathbb {Q} [\mathbb {Z} ])\simeq \operatorname {Hom} _{\mathbb {Q} [\mathbb {Z} ]}(H_{1}(X;\mathbb {Q} ),[\mathbb {Q} [\mathbb {Z} ]]/\mathbb {Q} [\mathbb {Z} ])$, 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.