Alexander Polynomial

The Alexander polynomial is a symmetric Laurent polynomial given by det(V - tV^t) 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 a_i,j the j^th free derivative of the i^th 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.