Alexander Invariant

The Alexander invariant H_*(X^~) of a knot K is the homology of the infinite cyclic cover of the complement of K, considered as a module over Lambda, the ring of integral laurent polynomials. The Alexander invariant for a classical tame knot is finitely presentable, and only H_1 is significant.

For any knot K^n in S^(n+2) whose complement has the homotopy type of a finite CW-complex, the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a tame knot in S^3 has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial denoted Delta(t).

See also

Alexander Ideal, Alexander Matrix, Alexander Polynomial

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Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206-207, 1976.

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Alexander Invariant

Cite this as:

Weisstein, Eric W. "Alexander Invariant." From MathWorld--A Wolfram Web Resource.

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