The dimer model, which is the study of perfect matchings on graphs, is ubiquitous in mathematical physics, exhibiting deep connections to algebraic geometry, statistical mechanics and quantum computation. In this talk, we first survey this landscape and then apply this rich theory to construct a unified framework for knot polynomials.

Focusing on the Alexander and Jones polynomials, we show that the computational distinction between them manifests in the topology of the underlying graph. We conclude by demonstrating how spin structures can be used to derive explicit determinantal formulas for these invariants, offering a geometric perspective on their structure.