On adjoint homological Selmer modules for SL$_2$-representations of knot groups

TL;DR

This paper introduces a new adjoint homological Selmer module for SL₂-representations of knot groups, demonstrating its torsion properties and providing concrete examples, thus drawing parallels with number theory conjectures.

Contribution

It defines the adjoint homological Selmer module for knot group representations and proves its finitely generated torsion-ness, a property linked to deep number theory conjectures.

Findings

Proves the torsion-ness of the adjoint Selmer module.

Provides concrete examples illustrating the module's properties.

Establishes a knot-theoretic analogue of a number theory concept.

Abstract

We introduce the adjoint homological Selmer module for an SL$_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for a Galois representation. We then show finitely generated torsion-ness of our adjoint Selmer module, which are widely known as conjectures in number theory, and give some concrete examples.