Knot signature functions are independent

A Seifert matrix is a square integral matrix satisfying

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

Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.