The Upsilon function of L-space knots is a Legendre transform

TL;DR

This paper demonstrates that for L-space knots, the Upsilon function can be expressed as a Legendre transform of a counting function related to d-invariants, linking knot concordance invariants through convex analysis.

Contribution

It establishes a new relationship between the Upsilon function and d-invariants for L-space knots, extending to connected sums and comparing their slice obstructions.

Findings

Upsilon function equals the Legendre transform of a d-invariant related counting function

Obstructions from Upsilon are contained within those from d-invariants for L-space knots

Generalizations show similar relationships hold for connected sums of L-space knots

Abstract

Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations apply for connected sums of L-space knots, which imply that the slice obstruction provided by Upsilon on the subgroup of concordance generated by L-space knots is no finer than that provided by the d-invariants.