Classifying spaces for knots: New bridges between knot theory and algebraic number theory

TL;DR

This paper explores the topological construction of classifying spaces in knot theory, revealing deep analogies with algebraic number theory, including connections between their fundamental groups and ideal class groups.

Contribution

It introduces explicit models of classifying spaces for knots and uncovers novel analogies linking knot theory to algebraic number theory.

Findings

Fundamental group of classifying spaces resembles ideal class groups.

A 9-term exact sequence for knots parallels Poitou-Tate sequence.

First (co)homology groups relate to Shafarevich groups in number fields.

Abstract

A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the framework of knot theory the notion of classifying spaces for families pops out naturally and we provide explicit models for them. The study of their topological properties hints at the existence of arcane connections between knot theory and algebraic number theory. Trying to track these connections, we discovered a behavioural analogy between the fundamental group of our classifying spaces and the ideal class group of an algebraic number field, a 9-term exact sequence for knots mimicking Poitou-Tate exact sequence for algebraic number fields and, in some cases, a second analogy between the first (co)homology group of the classifying spaces and the first Shafarevich group of a number field extension.