The determinant of a knot is det(V + V^{t}), where V is a Seifert Matrix for the knot.
Equivalently, this is Δ(-1), where Δ(t) is the Alexander polynomial.

The determinant is equal to the order of the homology of the two-fold branched cover of the knot.

**References**

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