A quantum N-dimer model

2025/10/08 by Douglas, Daniel C., Kenyon, Richard, Ovenhouse, Nicholas +2
#05E10 #57K31 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2510.07543

Abstract

We study a quantum version of the n-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in ℝ³). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in ℝ³, giving, in the planar setting, a quantum n-dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.

Citations

• Local Statistics of the M_n-Dimer Model
• Quantized six-vertex model on a torus
• Rank-N Dimer Models on Surfaces
• Monomial web basis for the SL(N) skein algebra of the twice punctured sphere
• 3d Quantum Trace Map
• Planar 3-webs and the boundary measurement matrix
• Higher-rank dimer models
• Higher Dimer Covers on Snake Graphs
• Rotation-invariant web bases from hourglass plabic graphs
• Dimers, webs, and local systems
• Stated SL(n)-Skein Modules and Algebras
• Gradient variational problems in ℝ²
• Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)
• From dimers to webs
• Tait colorings, and an instanton homology for webs and foams
• Dimers and cluster integrable systems
• Conformal invariance of loops in the double-dimer model
• Limit shapes and the complex burgers equation
• Cluster ensembles, quantization and the dilogarithm
• Dimers and Amoebae
• Conformal invariance of domino tiling
• Sl_n-character varieties as spaces of graphs

Related

• Dimer Models, Free Fermions and Super Quantum Mechanics
• Combinatorics of the Dimer Model on a Strip
• Topologically ordered phase states: from knots and braids to quantum dimers
• Conformal invariance of loops in the double-dimer model
• The Breaking of Geometric Constraint of Classical Dimers on the Square Lattice