Computing the knot signature from the Fox coloring (crossing) matrix

Derive a method to compute the classical signature of a knot from the Fox coloring matrix (also called the crossing matrix) of the universal Fox coloring group, allowing for appropriate correcting factors as needed.

Background

The knot signature is a classical integer-valued invariant central to low-dimensional topology and knot theory, an area in which Richard Litherland made significant contributions. Fox coloring matrices encode coloring constraints derived from a knot diagram and are connected to the universal Fox coloring group. This question asks whether the signature can be obtained directly from such a matrix, perhaps with correcting factors, providing a bridge between algebraic coloring data and the analytical signature invariant.