Representations of the (-2,3,7)-Pretzel Knot and Orderability of Dehn Surgeries

TL;DR

This paper constructs a family of SL(2,R) representations for the (-2,3,7)-pretzel knot and shows that certain Dehn surgeries on it are left-orderable, extending to other knots.

Contribution

It introduces a new family of representations for the pretzel knot and establishes orderability results for a range of Dehn surgeries, advancing understanding of knot group orderability.

Findings

Dehn surgeries with slopes less than 6 are left-orderable.

Constructed a 1-parameter family of SL(2,R) representations.

Extended orderability results to other knot examples.

Abstract

We construct a 1-parameter family of $\mathrm{SL}_2(\mathbf{R})$ representations of the pretzel knot $P(-2,3,7)$. As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples.