Topology Seminar 2008

Date: 10/21/08

Speaker: Eriko Hironaka

Title: Alexander polynomials

Abstract: Alexander polynomials are invariants of finitely presented groups up to second commutator subgroup. In particular, this polynomial invariant can be used to distinguish isotopy classes of knots and links in the three dimensional sphere. The polynomials are relatively simple to compute using Fox calculus, and are the determinant of a certain matrix with coefficients in the ring of Laurent polynomials. The matrix can be interpreted as a chain map on the simplicial homology of a cyclic covering of a suitably chosen CW-complex. For fibered links, or mapping tori, the matrix is also the monodromy of the mapping torus restricted to first homology. As a consequence, for fibered links the Alexander polynomial is monic.