Quantum statistical mechanics in arithmetic topology

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 33-sphere and cyclic branched coverings of the 33-sphere, which is an analog, in the sense of arithmetic topology, of the Bost–Connes system, with knots replacing primes, and cyclic branched coverings of the 33-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost–Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.