Quantum statistical mechanics in arithmetic topology

TL;DR

This paper constructs a quantum statistical mechanical system based on knots in the 3-sphere, drawing an analogy with number theory systems, and explores its algebraic properties and invariants.

Contribution

It introduces a novel quantum statistical mechanical model in arithmetic topology, replacing primes with knots and abelian extensions with cyclic branched coverings.

Findings

The system's operator algebra differs from Bost-Connes due to knot action properties.

The algebra of observables is a noncommutative Bernoulli crossed product.

Partition functions relate to simple knot invariants like genus and crossing number.

Abstract

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-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 3-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 crossed 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.