Knot signature functions are independent

Abstract

Seifert matrix is a square integral matrix V satisfying det( V - V-T) = +/-1. To such a matrix and unit complex number omega there corresponds a signature, sigma(omega)(V) = sign((1-omega) V + (1-(ω) over bar) V-T). Let S denote the set of unit complex numbers with positive imaginary part. We show that {sigma(omega)}(omegais an element ofS) 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)}(omegais an element ofA) is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot K subset of 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)}(omegais an element ofS) is linearly independent on the set of all knots and that the set of two{sided averaged signature functions, {sigma(omega)*}(omegais an element ofS), forms a linearly independent set of homomorphisms on the knot concordance group. Also, if nuis an element of 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 = (ν) over bar. We demonstrate that the results extend to the higher-dimensional setting.