On the topology of the coefficients of the Alexander-Conway polynomials of knots (Q2796948)

This paper focuses on the coefficient \(a_2\) of \(z^2\) in the Alexander-Conway polynomial \(\nabla_K(z)=a_{2m}z^{2m}+a_{2m-2}z^{2m-2}+\cdots+a_2z^2+a_0\) of a knot \(K\) (\(z=t+t^{-1}\)). The main result is that this coefficient can be expressed as the sum of the determinants of \(2\times 2\) sub-matrices of a particular Seifert matrix for \(K\). Since each of these determinants can be interpreted as a linking number of one part of the knot with another, the coefficient \(a_2\) can therefore be viewed as `a measure of self-linking of \(K\)'. The proof is elementary, making use of Taylor's Theorem and relationships between different entries of the Seifert matrix.