Left-orderable fundamental group and Dehn surgery on genus one two-bridge knots

TL;DR

This paper investigates the conditions under which Dehn surgery on genus one 2-bridge knots results in manifolds with left-orderable fundamental groups, depending on the surgery slope.

Contribution

It establishes a relationship between surgery slopes and left-orderability of the fundamental group for hyperbolic genus one 2-bridge knots.

Findings

Certain surgery slopes yield manifolds with left-orderable fundamental groups.

The range of slopes depends on the specific knot.

Provides new insights into the topology of 3-manifolds obtained from these knots.

Abstract

For any hyperbolic genus one 2-bridge knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ lies in some range which depends on the knot.