# 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.

## References

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

[2] Rolfsen, D., *Knots and Links,* AMS Chelsea Publishing, Providence (2003).

Privacy Notice