Knot Signature Functions Are Independent

A Seifert matrix is a square integral matrix V satisfying det(V − V T ) = ±1. To such a matrix and unit complex number ω there is a signature, σω(V ) = sign((1 − ω)V + (1 − ω̄)V T ). Let S denote the set of unit complex numbers with positive imaginary part. We show {σω}ω∈S is linearly independent, viewed as a set of functions on the set of all Seifert matrices. If V is metabolic, then σω(V ) = 0 unless ω is a root of the Alexander polynomial, ∆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 {σω}ω∈A is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot K ⊂ S one can associate a Seifert matrix VK , and σω(VK) induces a knot invariant. Topological applications of our results include a proof that the set of functions {σω}ω∈S is linearly independent on the set of all knots and that the set of two–sided averaged signature functions, {σ ω}ω∈S , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if ν is the root of some Alexander polynomial, there is a slice knot K satisfying σω(K) 6= 0 if and only if ω = ν or ω = ν̄. We demonstrate that the results extend to the higher dimensional setting.