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 alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has $M$-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.
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
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...
We study the Gukov--Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin--Morton--Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yiel...
Davide Passaro, Lara San Martín Suárez· 1 citation
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...
We formulate a precise conjecture relating integral form partially-symmetric Macdonald polynomials and the parabolic flag Hilbert schemes of Carlsson, Gorsky, and Mellit. This extends, in an explicit fashion, Haiman's realization of modified Macdonald symmetric functions via Hilbert schemes of points in the plane. As e...
Ben Goodberry, Daniel Orr· Combinatorial Theory· 4 citations
Let $T_{m,n}$ be the tadpole graph obtained by joining a cycle $C_m$ to a path $P_n$ by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fix...
Hoa Quang Tran, P. Nguyễn, Son Nguyen Thanh Tran· 0 citations
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