Trapezoidality of flat arrangement polynomials

Yuan Gao; Tianyu Yuan

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arXiv · 2608.21855

Trapezoidality of flat arrangement polynomials

Yuan Gao·Tianyu Yuan

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Abstract

Fox's conjecture predicts that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. K\'alm\'an, M\'esz\'aros, and Postnikov gave a new proof of trapezoidality for special alternating links using a polynomial associated with flat vector arrangements. We prove that the flat arrangement polynomial has trapezoidal coefficients for every full-dimensional flat vector arrangement, answering the corresponding open question. Our proof gives a geometric interpretation of this polynomial in terms of convex polytopes. As an application, we prove that the Murasugi--Stoimenow polynomial of every connected Eulerian digraph has trapezoidal coefficients.

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Yuan Gao, Tianyu Yuan. 2026-08-22. Trapezoidality of flat arrangement polynomials. https://arxiv.org/abs/2608.21855

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