Signature of a knot

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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 given by taking the linking number lk(a ⁺ ,b ⁻ ) where 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 b₁,...,b_2g for H₁(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 = φ(b_i,b_j). The signature of the matrix , thought of as a symmetric bilinear form, is the signature of the knot K.

Slice knots are known to have zero signature.

[edit] See also

• Link concordance