Aug 2026· 1 citation· ⚡ 1 influential· 15 references
Mathematics
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.
We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof...
We define an invariant of alternating links---a homogeneous, four-variable Laurent polynomial---that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special altern...
We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over $\mathbb F_q$, valid for every prime power $q$. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection tran...
Fox's conjecture famously asserts that the absolute values of the coefficients of the Alexander polynomial of alternating links are trapezoidal. In the setting of MOY graphs, where a different notion of Alexander polynomial appears, the equivalent result to Fox's conjecture is known to hold. In this paper, we relate th...
Leonardo Rodrigues de Medeiros, Arker Oke Soe· 0 citations
We prove Stieltjes'common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two $q\ge2$, the edge of slope $1/q$ in its $2$-adic Newton polygon has horizontal length less than $3q$. If a common nonzero roo...
We study sums associated with the action of $PGL_2(\mathbb Z)$ on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under su...
Dragomir Grozev, Navid Safaei· 0 citations
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