JSJ decompositions of knot exteriors, Dehn surgery and the $L$-space conjecture

TL;DR

This paper investigates the structure of 3-manifolds obtained by Dehn surgery on knots, using slope detection techniques, and explores implications for the $L$-space conjecture, including properties of fundamental groups and taut foliations.

Contribution

It introduces new methods to analyze JSJ decompositions and fundamental group properties of manifolds from knot surgeries, providing new proofs and extending results related to the $L$-space conjecture.

Findings

JSJ graph of knot exteriors is a rooted interval for certain surgeries.

Rational surgeries on composite knots have left-orderable fundamental groups.

Results connect $L$-space knots, Dehn surgeries, and taut foliations.

Abstract

In this article, we apply slope detection techniques to study properties of toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space knot or admits an irreducible rational surgery with non-left-orderable fundamental group, then the JSJ graph of its exterior is a rooted interval. Consequently, any rational surgery on a composite knot has a left-orderable fundamental group. This is the left-orderable counterpart of Krcatovich's result on the primeness of $L$-space knots, which we reprove using our methods. Analogous results on the existence of co-orientable taut foliations are proved when the knot has a fibred companion. Our results suggest a new approach to establishing the counterpart of Krcatovich's result for surgeries with co-orientable taut foliations, on which partial results have been achieved by Delman and Roberts. Finally, we prove results on left-orderable $p/q$-surgeries on knots with $p$ small.