Concordances from differences of torus knots to $L$-space knots

TL;DR

This paper investigates the structure of the concordance group generated by torus knots, showing that only the torus knots themselves are concordant to $L$-space knots within certain subgroups, thus clarifying their relationship.

Contribution

It proves that the subgroup generated by two positive torus knots contains no nontrivial $L$-space knots besides the torus knots, extending understanding of their concordance properties.

Findings

Subgroup generated by two positive torus knots contains no nontrivial $L$-space knots.

Connected sums of positive torus knots are not concordant to $L$-space knots.

Generalizations to larger subgroups are also discussed.

Abstract

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.