On L-space knots obtained from unknotting arcs in alternating diagrams

TL;DR

This paper investigates L-space knots derived from unknotting arcs in alternating diagrams, characterizing their types and showing finiteness results for knots with bounded genus within this class.

Contribution

It provides a classification of when these knots are torus, satellite, or hyperbolic, and proves finiteness of such knots with genus below a given threshold.

Findings

Characterization of when $K_D$ is a torus, satellite, or hyperbolic knot.

Finiteness of L-space knots in $\\mathcal D$ with genus less than $n$.

Identification of conditions for $K_D$ to be an L-space knot from alternating diagrams.

Abstract

Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by $\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given $n>0$, there are only finitely many L-space knots in $\mathcal D$ with genus less than $n$.