Alexander Polynomial


The Alexander polynomial is a symmetric Laurent Polynomial given by det(V - tVt) where V is a Seifert Matrix for the knot.

Alternatively, the Alexander polynomial can be calculated from a presentation of the knot group. It is the determinant of a generator for the first elementary ideal of the matrix A with entry ai,j the jth free derivative of the ith relation of the presentation.

The three genus is bounded below by half the degree of the Alexander Polynomial. The polynomial also provides insight for finding the concordance genus, the topological 4-genus, and the smooth 4-genus.


References

R. H. Fox, A Quick trip Through Knot Theory, Topology on 3-Manifolds, Prentice-Hall (1962) 120-167.

D. Rolfsen, Knots and Links, AMS Chelsea Publishing, Providence (2003)