Christoffel words and the strong Fox conjecture for two-bridge knots

Abstract

The trapezoidal Fox conjecture states that the coefficient sequence of the Alexander polynomial of an alternating knot is unimodal. We are motivated by a harder question, the strong Fox conjecture, which asks whether the coefficient sequence of the Alexander polynomial of alternating knots is actually log-concave. Our approach is to introduce a polynomial $\Delta(t)$ associated to a Christoffel word and to prove that its coefficient sequence is log-concave. This implies the strong Fox conjecture for two-bridge knots.