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

Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to [Formula: see text]-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 [Formula: see text]-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 [Formula: see text]-space knot.