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Trapezodial property of the generalized Alexander polynomial

Tamás Kálmán, Karola Mészáros, Alexander Postnikov

TL;DR

The paper develops a broad combinatorial framework that links the Alexander polynomials of special alternating links to a family of polynomials $f_A(t)$ defined from flat vector configurations. By embedding these polynomials into three matrix- and matroid-derived regimes (graphic, cographic, and totally positive), it simultaneously explains known trapezoidal/log-concave behavior and yields new proofs, including Fox's conjecture for special alternating links. Key tools include external semi-activity on oriented matroids, zonotopal tilings, and Lorentzian polynomial theory, which together give refined lattice-point interpretations and box-positivity properties. The results unify knot invariants with polyhedral geometry and matroid theory, offering a versatile approach to trapezoidal coefficient structures in broader classes of polynomials and potentially guiding future extensions to non-special alternating links.

Abstract

Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this date. Recently Fox's conjecture was settled for all special alternating links. In this paper we take a broad view of the Alexander polynomials of special alternating links, showing that they are a generating function for a statistic on certain vector configurations. We study three types of vector configurations: (1) vectors arising from cographic matroids, (2) vectors arising from graphic matroids, (3) vectors arising from totally positive matrices. We prove that Alexander polynomials of special alternating links belong to both classes (1) and (2), and prove log-concavity, respectively trapezoidal, properties for classes (2) and (3). As a special case of our results, we obtain a new proof of Fox's conjecture for special alternating links.

Trapezodial property of the generalized Alexander polynomial

TL;DR

The paper develops a broad combinatorial framework that links the Alexander polynomials of special alternating links to a family of polynomials defined from flat vector configurations. By embedding these polynomials into three matrix- and matroid-derived regimes (graphic, cographic, and totally positive), it simultaneously explains known trapezoidal/log-concave behavior and yields new proofs, including Fox's conjecture for special alternating links. Key tools include external semi-activity on oriented matroids, zonotopal tilings, and Lorentzian polynomial theory, which together give refined lattice-point interpretations and box-positivity properties. The results unify knot invariants with polyhedral geometry and matroid theory, offering a versatile approach to trapezoidal coefficient structures in broader classes of polynomials and potentially guiding future extensions to non-special alternating links.

Abstract

Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this date. Recently Fox's conjecture was settled for all special alternating links. In this paper we take a broad view of the Alexander polynomials of special alternating links, showing that they are a generating function for a statistic on certain vector configurations. We study three types of vector configurations: (1) vectors arising from cographic matroids, (2) vectors arising from graphic matroids, (3) vectors arising from totally positive matrices. We prove that Alexander polynomials of special alternating links belong to both classes (1) and (2), and prove log-concavity, respectively trapezoidal, properties for classes (2) and (3). As a special case of our results, we obtain a new proof of Fox's conjecture for special alternating links.
Paper Structure (28 sections, 40 theorems, 33 equations, 5 figures)

This paper contains 28 sections, 40 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Knot theory background
  3. Fox's trapezoidal conjecture
  4. Special alternating links
  5. The Alexander polynomial in the case of special alternating links
  6. Polynomials arising from matrices
  7. Oriented matroids arising from matrices
  8. External semi-activity of the bases of an oriented matroid
  9. Volume form and unimodular arrangements
  10. Definition and invariance of the polynomials $f_{{A}}(t)$
  11. Flat matrices arising from graphic and cographic matroids
  12. Flat graphic matroids
  13. Flat cographic matroids
  14. The planar bipartite and Eulerian case
  15. The Murasugi--Stoimenow polynomials $P_D$ as $f_{M_D^*}$
  16. ...and 13 more sections

Key Result

Theorem A

The Alexander polynomial $\Delta_L(t)$, associated to a special alternating link $L$, equals $f_{{A}}(-t)$ for a flat matrix $A$, whose oriented matroid $M[A]$ is both graphic and cographic.

Figures (5)

  • Figure 1: A plane bipartite graph $G$, medial graph $M(G)$, the associated special alternating link $L_G$, and the dual graph $G^*$ of $G$ endowed with an Eulerian orientation that makes it into an alternating dimap $D$.
  • Figure 2: On the left is a directed graph $D$. On the right is its signed incidence matrix $I(D)$.
  • Figure 3: On the left is a directed graph $D$ along with a spanning tree $T$. On the right is the matrix $A_{D,T}$ of Definition \ref{['def:graph']}, containing the highlighted block $M$.
  • Figure 4: On the left is a directed graph $D$ along with a fixed spanning tree $T$. On the right is the matrix $B_{D,T}$ of Definition \ref{['def:cograph']}, with its block $K$ highlighted.
  • Figure 5: The flow values on the edges of $\vec{G}$ with netflow vector $l$ coincide with the coefficients $\alpha_j$ from Definition \ref{['def:adm']}.

Theorems & Definitions (80)

  • Theorem A
  • Theorem B
  • Corollary
  • Theorem C
  • Conjecture 2.1
  • Example 2.3
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • ...and 70 more