The ϒ function ofL–space knots is a Legendre transform

Abstract

Given an L-space knot we show that its Υ function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Υ 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 Υ on the subgroup of concordance generated by L-space knots is no finer than that provided by the d-invariants.

Heegaard Floer homology and concordance bounds on the Thurston norm

“…If a knot K can be unknotted by a single generalised crossing change such that the untwisting unknot has surgery coefficient −1 we will say that it has negative untwisting number 1. Note that this generalises the corresponding property for knots with unknotting number 1 (see [3,Theorem 6.1]). This should also be compared with the behaviour of Ozsváth-Szabó's τ under twisting (see [13]).…”

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

“…We compute that L 4 5 ,s has slope m = − 3 2 and j-intercept b = 5s 2 . In Figure 4, one can see that L 4 5 ,s with minimal s passes through the points (1,8) and (3,5). The j-intercept of this line is 19 2 corresponding to an s value of 19 5 .…”

“…Thus γ T (5,7) ( 4 5 ) = 19 5 . Note that near t = 4 5 , the line L t,s pivots around the two points (1,8) and (3,5). This causes a change in slope in Υ K and so t = 4 5 is a singulariy of Υ ′ K .…”

“…Now, we turn our attention to secondary Upsilon. We have that ( 4 5 ), we compute how far the line of slope − 3 2 needs to be moved so that the elements represented by (1,8) and (3,5) are homologous in F 4 5 ,r . In the diagram we see that we need F 4 5 ,r to contain the elements represented by (3,7) and (2,8).…”

“…Because of their role in studying algebraic curves, research on invariants of torus knots continues. In particular, the Ozsváth-Stipsicz-Szabó Upsilon function has been used in [1] and [3]. Recently, Feller and Krcatovich (in [4]) determined relationships among the Upsilon functions of torus knots.…”