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

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.

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

“…This observation was examined further in Livingston [9] who established that a nontrivial connected sum of positive torus knots K is concordant to an L-space knot if and only if K is a single torus knot by using the Levine-Tristam signature function [3] [6] [14] and the tau invariant [11] [13]. Then Allen [2] proved that no linear combination of two torus knots is concordant to an L-space knot as well as making steps towards the general case by using the tau invariant, upsilon invariant [12], and the Levine-Tristram signature function. This previous work led to the following conjecture.…”

“…Conjecture (Allen [2]). If a connected sum of (possibly several) torus knots is concordant to an L-space knot, then it is concordant to a positive torus knot.…”

“…We shall begin by introducing theorems of Livingston [9] and Allen [2]. The following proposition, found in Livingston [9], proves Allen's conjecture for positive torus knots.…”

“…Allen [2] established the following proposition relating the Alexander polynomial of a linear combination of torus knots to a concordant L-space knot. Proposition 2.2 (Allen [2]).…”