Knot signature functions are independent

Abstract:

A Seifert matrix is a square integral matrix $V$ satisfying \begin{equation*}\det (V - V^T) =\pm 1. \end{equation*} To such a matrix and unit complex number $\omega$ there corresponds a signature, \begin{equation*}\sigma _\omega (V) = \mbox {sign}( (1 - \omega )V + (1 - \bar {\omega })V^T). \end{equation*} Let $S$ denote the set of unit complex numbers with positive imaginary part. We show that $\{\sigma _\omega \}_ { \omega \in S }$ is linearly independent, viewed as a set of functions on the set of all Seifert matrices. If $V$ is metabolic, then $\sigma _\omega (V) = 0$ unless $\omega$ is a root of the Alexander polynomial, $\Delta _V(t) = \det (V - tV^T)$. Let $A$ denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that $\{\sigma _\omega \}_ { \omega \in A }$ is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot $K \subset S^3$ one can associate a Seifert matrix $V_K$, and $\sigma _\omega (V_K)$ induces a knot invariant. Topological applications of our results include a proof that the set of functions $\{\sigma _\omega \}_ { \omega \in S }$ is linearly independent on the set of all knots and that the set of two–sided averaged signature functions, $\{\sigma ^*_\omega \}_ { \omega \in S }$, forms a linearly independent set of homomorphisms on the knot concordance group. Also, if $\nu \in S$ is the root of some Alexander polynomial, then there is a slice knot $K$ whose signature function $\sigma _\omega (K)$ is nontrivial only at $\omega = \nu$ and $\omega = \overline {\nu }$. We demonstrate that the results extend to the higher-dimensional setting.