Infinite families of non-left-orderable L-spaces

TL;DR

This paper constructs infinite families of non-left-orderable L-spaces from alternating tangles, supporting a conjecture linking L-spaces and non-left-orderability, and provides explicit descriptions and examples including hyperbolic and Seifert fibered spaces.

Contribution

It introduces a method to generate infinite families of non-left-orderable L-spaces from alternating tangles, expanding understanding of their structure and properties.

Findings

Constructed infinite families of non-left-orderable L-spaces from alternating tangles.

Provided explicit descriptions as double branched coverings over hyperbolic links.

Included many non-Seifert fibered and Seifert fibered spaces within these families.

Abstract

For each connected alternating tangle, we provide an infinite family of non-left-orderable L-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational homology 3-sphere is an L-space if and only if it is non-left-orderable. These 3-manifolds are obtained as Dehn fillings of the double branched covering of any alternating encircled tangle. We give a presentation of these non-left-orderable L-spaces as double branched coverings of S^3, branched over some specified links that turn out to be hyperbolic. We show that the obtained families include many non-Seifert fibered spaces. We also show that these families include many Seifert fibered spaces and give a surgery description for some of them. In the process we give another way to prove that the torus knots T(2, 2m+1) are L-space-knots as has already been shown by Ozsv\'ath and Szab\'o in [24].