Remark on arithmetic topology

Abstract

We formalize the arithmetic topology, i.e. a relationship between knots and primes. Namely, using the notion of a cluster C*-algebra we construct a functor from the category of 3-dimensional manifolds M to a category of algebraic number fields K, such that the prime ideals (ideals, resp.) in the ring of integers of K correspond to knots (links, resp.) in M. It is proved that the functor realizes all axioms of the arithmetic topology conjectured in the 1960's by Manin, Mazur and Mumford.

10 pages, 2 figures; reference to the Lickorish-Wallace Theorem is added

References in corpus (2)

• On cluster C*-algebras
• Cluster C*-algebras and knot polynomials

Cited by in corpus (3)

• Arithmetic topology of 4-manifolds
• Dynamical ideals of non-commutative rings
• Geometry of integers revisited