Usbrth

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

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

beginalign
sigma(K_+)−sigma(K_-)in0,2tagalabelprop-a\
4|sigma(𝐾)⟺∇_K(2𝑖)>0
tagblabelprop-b
endalign

and then to observe that eqrefprop-a and eqrefprop-b together uniquely determine the number $sigma(K)$.

Namely, pick a sequence of crossing changes $K=K_varepsilon_0rightarrow K_varepsilon_0varepsilon_1 rightarrow K_varepsilon_0varepsilon_1varepsilon_2rightarrow dotsrightarrow K_varepsilon_0varepsilon_1dotsvarepsilon_N=U$ to the unknot. Use eqrefprop-b to calculate $sigma(K_varepsilon_1dotsvarepsilon_m)pmod 4$ for each $mgeq 0$. 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 eqrefprop-b of the knot signature is proven.