Knot signature functions are independent

Abstract

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