Remark on arithmetic topology

2017/06/20 by Nikolaev, Igor
#11R04 #46L85 #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1706.06398

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.

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