Signature of a Knot
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High Quality Content by WIKIPEDIA articles! 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 b1,...,b2g for H1(S) (where g is the genus of the surface) the Seifert form can b...
High Quality Content by WIKIPEDIA articles! 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 b1,...,b2g 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, Vij = (bi,bj). The signature of the matrix V+V^perp, thought of as a symmetric bilinear form, is the signature of the knot K. Slice knots are known to have zero signature.
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