On the universal deformations for SL_2-representations of knot groups

TL;DR

This paper develops a deformation theory for SL_2-representations of knot groups, establishing the existence of universal deformations, connecting with character schemes, and exploring explicit examples and geometric implications.

Contribution

It introduces a universal deformation framework for SL_2-representations of knot groups, extending analogies with Galois deformation theory and linking to hyperbolic geometry.

Findings

Proves existence of universal deformations for SL_2-representations.

Connects deformations with character schemes for algebraically closed fields.

Provides explicit universal deformations for Riley representations and hyperbolic knot groups.

Abstract

Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given SL_2-representation of a finitely generated group Pi over a field whose characteristic is not 2. We then show its connection with the character scheme for SL_2-representations of Pi when k is an algebraically closed field. We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations. Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston's theory on deformations of hyperbolic structures.