A dimer view on Fox's trapezoidal conjecture

Fox’s conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been settled, some quite recently: Fox’s conjecture was shown to hold for special alternating links by Hafner, Mészáros, and Vidinas (2023) and for certain diagrammatic Murasugi sums of special alternating links by Azarpendar, Juhász, and Kálmán (2024). In this paper, we give an alternative proof of Azarpendar, Juhász, and Kálmán’s aforementioned beautiful result via a dimer model for the Alexander polynomial. In doing so, we not only obtain a significantly shorter proof of Azarpendar, Juhász, and Kálmán’s result than the original, but we also obtain several theorems of independent interest regarding the Alexander polynomial, which are readily visible from the dimer point of view.

💡 Research Summary

This paper provides a novel combinatorial perspective on Fox’s longstanding trapezoidal conjecture in knot theory by employing a dimer model (perfect matching) framework for the Alexander polynomial.

The work begins by revisiting Kauffman’s state sum formula for the (symmetrized) Alexander polynomial of a link diagram. The authors then construct the “face-crossing incidence graph” of the diagram and its “truncated” version relative to a chosen segment. By assigning specific edge weights derived from the local crossing weights in Kauffman’s formula, they establish a weight-preserving bijection between Kauffman states and perfect matchings on this truncated graph. Consequently, the Alexander polynomial Δ̃_L is precisely the generating function for weighted perfect matchings (dimers) on the graph G_L,i.

The core of the analysis focuses on the structure of alternating links. A link diagram is “special alternating” if all its Seifert circles (obtained by smoothing crossings) are of type 1 (bounding a region). A general alternating link diagram can be decomposed along a type 2 Seifert circle into a “diagrammatic Murasugi sum” of two smaller diagrams, L’ and L’’. The authors translate this topological decomposition into the language of the dimer graph G_L,i. The graph contains a specific subgraph, called a “flock,” associated with the type 2 circle C, which naturally separates into components corresponding to G_L’,i’ and G_L’’,i’'.

Leveraging this graphical decomposition, the authors prove several key theorems. First, they establish coefficient-wise inequalities for the absolute values of the Alexander polynomials under Murasugi sum (Theorem 1.3). For a sum along a circle of length 1, |Δ̃_L| = |Δ̃_L’|·|Δ̃_L’’|. For length ≥ 2, |Δ̃_L| is bounded below by |Δ̃_L’|·|Δ̃_L’’| + |Δ̃_Ł’|·|Δ̃_Ł’’|, where Ł’ and Ł’’ are specific modifications of L’ and L’’. Second, they prove that Δ̃_L and the product Δ̃_L’·Δ̃_L’’ have identical support, meaning they have nonzero coefficients for the exact same set of monomials (Theorem 1.4).

The primary application of this framework is a new, significantly shorter proof of a recent result by Azarpendar, Juhász, and Kálmán (Theorem 1.2). Their result states that Fox’s conjecture holds for alternating link diagrams whose type 2 Seifert circles all have “length” at most 2. The dimer model proof elegantly combines the graphical decomposition, the aforementioned inequalities, and the known trapezoidality for special alternating links (the base case) to establish the result. The proof strategy is clearer and avoids many technical complexities of the original.

Finally, the paper concludes with computational evidence and discussion explaining why the current dimer-based approach seems to hit a barrier when a type 2 Seifert circle has length 3 or more, thus pinpointing the obstacle that must be overcome to fully resolve Fox’s conjecture using these methods. The work not only simplifies a recent advance but also enriches the understanding of the Alexander polynomial through a powerful combinatorial lens.