On certain L-functions for deformations of knot group representations

TL;DR

This paper explores the properties of an $L$-function associated with deformations of knot group representations, demonstrating torsion properties and zeroes' simplicity through theoretical and example-based analysis.

Contribution

It introduces an $L$-function for deformations of knot group representations and investigates its properties, addressing two open problems posed by Mazur.

Findings

Proves the torsion property of the twisted knot module under certain conditions.

Verifies the simplicity of zeroes of the $L$-function in specific 2-bridge knot examples.

Abstract

We study the twisted knot module for the universal deformation of an ${\rm SL}_2$-representation of a knot group, and introduce an associated $L$-function, which may be seen as an analogue of the algebraic $p$-adic $L$-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we verify the simplicity of the zeroes of the $L$-function by some concrete examples for 2-bridge knots.