We prove infinitely many cases of conjectured sharp upper and lower bounds for the spanning tree entropy of any planar lattice graph. These bounds come from volumes of associated hyperbolic alternating links, right-angled hyperbolic polyhedra and hyperbolic regular ideal bipyramids. For many planar lattice graphs, we show these bounds are easy to compute and provide excellent numerical estimates for the spanning tree entropy.

The research of both authors is partially supported by grants from the Simons Foundation and PSC-CUNY.