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.

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