Reference request for knot's signature via skein relation

Question

Mathworld's article on knot signature [1] defines it as a function that satisfies two conditions: $s(K_+) - s(K_-) \in \{0, 2\}$ and $4 \mid s(K) \iff \nabla (K)(2i) > 0$. Where can I find proof that this is in fact equivalent to classical definition as signature of matrix $M + M^T$, where $M$ is a Seifert matrix?

[1] http://mathworld.wolfram.com/KnotSignature.html

Answer 1

The idea is to prove that the standard definition $$\sigma(K)=sgn(M+M^T)$$ implies the two properties:

\begin{align} \sigma(K_+)−\sigma(K_-)\in\{0,2\}\tag{a}\label{prop-a}\\ 4|\sigma()⟺∇_K(2)>0 \tag{b}\label{prop-b} \end{align}

and then to observe that \eqref{prop-a} and \eqref{prop-b} together uniquely determine the number $\sigma(K)$.

Namely, pick a sequence of crossing changes $K=K_{\varepsilon_0}\rightarrow K_{\varepsilon_0\varepsilon_1} \rightarrow K_{\varepsilon_0\varepsilon_1\varepsilon_2}\rightarrow \dots\rightarrow K_{\varepsilon_0\varepsilon_1\dots\varepsilon_N}=U$ to the unknot. Use \eqref{prop-b} to calculate $\sigma(K_{\varepsilon_1\dots\varepsilon_m})\pmod 4$ for each $m\geq 0$. This will allow you to determine if after each crossing change the signature stays the same or changes by 2. Since $\sigma(U)=0$, you can recover $\sigma(K)$ by adding up all the 2's that occurred.

This is explained in the Remark 3 (p. 97) in:

Giller, C. (1982). A Family of Links and the Conway Calculus. Transactions of the American Mathematical Society, 270(1), 75-109.

relying on the Theorem 5.6 (p. 399) in:

Murasugi, K. (1965). On a certain numerical invariant of link types. Transactions of the American Mathematical Society, 117, 387-422.

where the property \eqref{prop-b} of the knot signature is proven.