Concordances of sums of alternating torus knots and their mirrors to $L$-space knots

TL;DR

This paper investigates when linear combinations of torus knots are concordant to $L$-space knots, proving a conjecture for alternating torus knots and establishing necessary conditions for general cases.

Contribution

It proves Allen's conjecture for alternating torus knots and provides necessary conditions for linear combinations of torus knots to be concordant to $L$-space knots.

Findings

Proved that a linear combination of alternating torus knots is concordant to an $L$-space knot iff it is a single torus knot.

Established a necessary condition for linear combinations of torus knots to be concordant to $L$-space knots.

Extended the understanding of concordance relations among torus knots and $L$-space knots.

Abstract

Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to $L$-space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear combination of alternating torus knots is concordant to an $L$-space knot if and only if the connected sum is a single torus knot. Then we establish a necessary condition for when a linear combination of torus knots is concordant to an $L$-space knot.