Efficient cooling for the unknot’s Hodge Laplacian

Prove or refute that for any planar diagram of the unknot, the Hodge Laplacian of its Khovanov complex can be cooled (e.g., via Gibbs sampling or an efficiently thermalizing Lindbladian) in polynomial time to a temperature that yields inverse-polynomial overlap with the two-dimensional ground space, enabling efficient verification of unknottedness.

Background

The proposed quantum algorithm relies on preparing a low-temperature Gibbs state of the Hodge Laplacian and then projecting onto its kernel. For the unknot, the kernel is known to be two-dimensional. If efficient cooling is possible for any representation of the unknot, the algorithm would yield an efficient unknot verification method. Establishing or refuting this conjecture requires understanding thermalization properties tied to the Laplacian’s spectrum.