Remark on arithmetic topology
Geometric Topology
2017-12-27 v2 Number Theory
Operator Algebras
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.
Cite
@article{arxiv.1706.06398,
title = {Remark on arithmetic topology},
author = {Igor Nikolaev},
journal= {arXiv preprint arXiv:1706.06398},
year = {2017}
}
Comments
10 pages, 2 figures; reference to the Lickorish-Wallace Theorem is added