On Alexander polynomial of a knot

Question

The Alexander polynomial of a knot is of the form $$\Delta(t)=det(V^T-tV),$$ where $V$ is the Seifert matrix, see http://archive.lib.msu.edu/crcmath/math/math/a/a116.htm. What is geometric or some other meaning of the zeros of the polynomial? I've encountered a similar expression in the theory of 2D/planar electrical networks, where the zeros are essentially multipliers of harmonic continuation on the networks.

Answer 1

The zeros of the Alexander polynomial are the values where the Tristam-Levine signature of the knot can jump. Let $V$ be a Seifert matrix of $K$ and let $\omega\in\mathbb{C}$ be on the unit circle. The Tristam-Levine signature $\sigma_\omega(K)$ is defined as the signature (number of positive eigenvalues minus number of negative eigenvalues) of the matrix $$(1-\omega)V + (1-\overline{\omega})V^T.$$ The Tristam-Levine signature is locally constant away from zeros of the Alexander polynomial, but can jump from one value to another on either side of a zero.

On another note, the zeros of the Alexander polynomial can say something about the growth rates of the first homology of the cyclic branched covers of the knot. Let $\Sigma_n(K)$ be the $n$-fold cyclic branched cover of $S^3$ branched along $K$. Furthermore, let $b_n=|H_1(\Sigma_n(K))|$ be the size of the first homology of the $n$-fold branched cover. Gordon proved that all of the zeros of the Alexander polynomial of $K$ are roots of unity if and only if the sequence $\{b_n\}$ is periodic. Gonzalez-Acuna and Short proved the if the Alexander polynomial has a zero that is not a root of unity, then the sequence $\{b_n\}$ tends to infinity.

Finally, Hoste conjectured that if $K$ is an alternating knot, then the real part of any zero $\alpha$ of the Alexander polynomial of $K$ satisfies $\operatorname{Re}(\alpha)>-1.$ Lyubich and Murasugi proved Hoste's conjecture in some special cases. The general case is still open. Stoimenow showed that Hoste's conjecture and the conjecture that the coefficients of the Alexander polynomial of an alternating knot form a log-concave sequence are essentially independent.