An involutive upsilon knot invariant

Abstract

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsváth-Stipsicz-Szabó. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot.

On the nonorientable four-ball genus of torus knots

“…Given a knot K in S 3 , we construct a module over the ring F 2 [U, Q]/ Q 2 , which we call the involutive unoriented knot Floer homology HF KI (K). Using the structure of HF KI (K), we can define two concordance invariants ῡ and υ ¯, which are the values at 1 of the involutive Upsilon invariants Ῡt and Υ ¯t defined by Hogancamp and Livingston [HL17]. Although HF KI itself does not seem to satisfy the strong functoriality properties that Heegaard Floer theory and its variants satisfy, we can still show that ῡ and υ ¯behave nicely under nonorientable cobordisms.…”

Unoriented Cobordism Maps on Link Floer Homology

“…Further developments and possible applications. As an application of the disoriented link cobordism theory, we will extend the involutive upsilon invariant defined by Hogancamp and Livingston [HL17] from knots to links. Furthermore, we will study the relation between involutive upsilon invariant and the unoriented four-ball genus for disoriented link cobordism.…”