The Beta Function of a Knot

In this note we study a metric invariant of a knot K embedded in R 3. This metric invariant depends on a complex parameter s. It is defined as a double integral using the parameterization x → γ(x) of K by the arc-length parameter x: B K (s) = K×K ||γ(x) − γ(y)|| s dxdy (1) where dx, dy denote the arc-lengths on the two copies of K. Clearly the function (x, y) → ||γ(x) − γ(y)|| s is continuous for Re(s) > 0, so that the integral converges in that domain. One main result is Theorem 0.1. For a smooth knot K, the function s → B K (s) extends analytically to a meromorphic function on C, with only possible poles at −1, −3, −5 · · ·. The residue at −2j − 1 is of the form K P j (κ, τ)dx, where P j is an explicitly computable polynomial in the curvature κ, the torsion τ and their derivatives.