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

Ian M. Banfield

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arXiv · 2212.04561

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

Ian M. Banfield

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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.

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BibTeX RIS

Ian M. Banfield. 2022-12-08. Christoffel words and the strong Fox conjecture for two-bridge knots. https://arxiv.org/abs/2212.04561

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