Traceless SU(2) representations of 2-stranded tangles

TL;DR

This paper studies traceless SU(2) representations of 2-stranded tangles in homology 3-balls, characterizing their structure, especially the binary dihedral subset, and analyzing the restriction map to the pillowcase, with applications to knots.

Contribution

It completely determines the binary dihedral representations and character varieties for many tangles related to knots, and analyzes the restriction map to the pillowcase, revealing linearity in certain cases.

Findings

Binary dihedral representations are fully characterized.

The restriction map's image can be non-linear, but is linear for tangles from pretzel knots.

The paper provides explicit examples and applications to knot theory.

Abstract

Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we investigate the character variety $R(Y,T)$ of conjugacy classes of traceless SU(2) representations of $\pi_1(Y\setminus T)$. In particular we completely determine the subspace of binary dihedral representations, and identify all of $R(Y,T)$ for many tangles naturally associated to knots in S^3. Moreover, we determine the image of the restriction map from $R(T,Y)$ to the traceless SU(2) character variety of the 4-punctured 2-sphere (the {\it pillowcase}). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.