[Paper Review] The Upsilon function of L-space knots is a Legendre transform

Maciej Borodzik, Matthew Hedden|arXiv (Cornell University)|May 25, 2015

Geometric and Algebraic Topology26 references4 citations

TL;DR

This paper establishes that the Υ function of an L-space knot is the Legendre transform of a counting function derived from its d-invariants under large surgeries. The key result shows that for L-space knots and their connected sums, the concordance obstruction from Υ is fully determined by the d-invariants, meaning Υ provides no finer information than the d-invariants in this setting.

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.

Motivation & Objective

• To clarify the relationship between the Υ function and d-invariants for L-space knots.
• To determine whether the concordance obstruction from Υ is strictly stronger than that from d-invariants in the context of L-space knots.
• To investigate whether the Legendre transform structure of Υ reflects deeper geometric or algebraic properties of L-space knots.
• To explore the implications of this relationship for estimating Gordian distance and crossing change obstructions.
• To examine the extent to which the Legendre transform characterization extends beyond L-space knots, particularly to strongly quasipositive knots.

Proposed method

• Define the function $ J(x) $ as a counting function derived from the knot Floer homology invariants of an L-space knot.
• Apply the Legendre transform to $ 2J(-x) $, showing it yields the Υ function via Theorem 1.1.
• Use the d-invariant obstruction from negative-definite 4-manifolds to derive inequalities for $ J $-functions under crossing changes.
• Establish that the crossing change inequalities for $ \Upsilon $ are implied by those for $ J $, via Legendre transform duality.
• Leverage the fact that the Legendre transform reverses inequalities and commutes with shifts to relate $ \Upsilon $ and $ J $ under cobordism.
• Use explicit computations on torus knots (e.g., $ T(4,9) $, $ T(6,7) $) to demonstrate that $ J $-inequalities can obstruct crossing changes where $ \Upsilon $ cannot.

Experimental results

Research questions

• RQ1 Is the Υ function of an L-space knot equal to the Legendre transform of $ 2J(-x) $, where $ J $ encodes d-invariants of large surgeries?
• RQ2 Does the $ d $-invariant obstruction for concordance and crossing changes contain strictly more information than the $ \Upsilon $ function for connected sums of L-space knots?
• RQ3 For which knots is $ \Upsilon(t) $ convex, and when is it the Legendre transform of $ 2J(-x) $?
• RQ4 Can the $ \Upsilon $-based crossing change obstruction be derived entirely from the $ J $-function in the case of L-space knots?
• RQ5 Do the $ \Upsilon $ and $ J $ functions provide equivalent or distinct information for the Gordian distance between algebraic knots?

Key findings

• The Υ function of any L-space knot is the Legendre transform of $ 2J(-x) $, where $ J(x) $ is a counting function derived from the $ d $-invariants of large surgeries.
• For connected sums of L-space knots, the concordance obstruction from $ \Upsilon $ is entirely contained within the $ d $-invariants, meaning $ \Upsilon $ provides no additional information in this subgroup.
• The crossing change inequality for $ \Upsilon $ is implied by the corresponding inequality for $ J $, but not conversely, demonstrating that $ J $-based obstructions are strictly stronger.
• An explicit example with $ T(4,9) $ and $ T(6,7) $ shows that $ \Upsilon $ cannot obstruct three positive crossing changes, but the $ J $-function does, proving their Gordian distance is at least four.
• The $ \Upsilon $ function of an L-space knot is not convex in general, but the Legendre transform structure reveals that its convexity is tied to the $ d $-invariant counting function.
• The result does not extend to all knots—$ \Upsilon $ is typically not convex, while the Legendre transform of a real function is always convex, indicating a fundamental limitation of the transform in general settings.