Dimer Model for All (p,q) Torus Knots

Establish whether every torus knot T(p,q)—the closure of the braid (∏_{i=1}^{p−1} σ_i)^q in the braid group B_p—admits a dimer model, meaning that for T(p,q) the partition function Z(T(p,q)) defined as the sum over perfect matchings of the balanced overlaid Tait graph (with each matching weighted by the product of activity-letter specializations on its edges) equals the bracket polynomial ⟨T(p,q)⟩.

Background

The paper proves that (2,q) torus knots and a family of links obtained as closures of homogeneous braids admit a dimer model, i.e., the sum over perfect matchings of the balanced overlaid Tait graph (equivalently, a determinant of a weighted adjacency matrix via a Kasteleyn weighting) recovers the bracket polynomial and hence the Jones polynomial.

Motivated by these results and the combinatorial efficiency of the determinant approach, the authors propose extending this methodology from (2,q) torus knots to all (p,q) torus knots, which are closures of (∏_{i=1}^{p−1} σ_i)^q in B_p. Establishing a dimer model for all torus knots would provide a polynomial-time algorithm for computing their Jones polynomial via the balanced overlaid Tait graph framework.