Seifert Form

For K a given knot in S^3, choose a Seifert surface M^2 in S^3 for K and a bicollar M^^×[-1,1] in S^3-K. If x in H_1(M^^) is represented by a 1-cycle in M^^, let x^+ denote the homology cycle carried by x×1 in the bicollar. Similarly, let x^- denote x×-1. The function f:H_1(M^^)×H_1(M^^)->Z defined by


where lk denotes the linking number, is called a Seifert form for K.

See also

Seifert Matrix

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-201, 1976.

Referenced on Wolfram|Alpha

Seifert Form

Cite this as:

Weisstein, Eric W. "Seifert Form." From MathWorld--A Wolfram Web Resource.

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