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

TL;DR

This paper proves the strong Fox conjecture for two-bridge knots by demonstrating that a polynomial associated with Christoffel words has a log-concave coefficient sequence, advancing understanding of Alexander polynomial properties.

Contribution

The paper introduces a polynomial linked to Christoffel words and proves its coefficient sequence is log-concave, confirming the strong Fox conjecture for two-bridge knots.

Findings

Proved log-concavity of the polynomial's coefficients

Confirmed the strong Fox conjecture for two-bridge knots

Connected Christoffel words with Alexander polynomial properties

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.