Knot signature functions are independent

A Seifert matrix is a square integral matrix $V$ satisfying

To such a matrix and unit complex number $\omega$ there corresponds a signature,

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.