Heegaard Floer Homology, The \(L\)-Space Conjecture, and Left-Orderable Surgeries on 2-Bridge Knots

Abstract

In this paper, we discuss the \(L\)-space conjecture, which hopes to establish a connection between the left-orderability of the fundamental group of a rational homology 3-sphere and its Heegaard Floer homology. We first present a review of Heegaard Floer homology and then a review of this conjecture itself, followed by a discussion of techniques used to solve special cases of this conjecture. We then find a new method for showing fundamental groups of certain Dehn surgeries on 2-bridge knots are left-orderable, and use this method to show that surgeries on the knot \(6_2\) with slopes in the interval \((-4, 8)\cap\mathbb{Q}\) are left-orderable. We additionally find an infinite family of 2-bridge knots for which surgeries with slopes in the interval \((-4, 4)\cap\mathbb{Q}\) are left-orderable.