An explicit description of $(1,1)$ L-space knots, and non-left-orderable surgeries

TL;DR

This paper provides an explicit description of (1,1) L-space knots in $S^3$ and lens spaces, and proves that surgeries on these knots produce manifolds with non-left-orderable fundamental groups.

Contribution

It offers a new explicit characterization of (1,1) L-space knots and establishes a universal property of their Dehn surgeries regarding fundamental group orderability.

Findings

Explicit description of (1,1) L-space knots

Surgeries yield non-left-orderable fundamental groups

Extension of previous characterizations

Abstract

Greene, Lewallen and Vafaee characterized $(1,1)$ L-space knots in $S^3$ and lens space in the notation of coherent reduced $(1,1)$-diagrams. We analyze these diagrams, and deduce an explicit description of these knots. With the new description, we prove that any L-space obtained by Dehn surgery on a $(1,1)$-knot in $S^3$ has non-left-orderable fundamental group.