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.

**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)