Upsilon-type concordance invariants

Abstract

To a region $C$ of the plane satisfying a suitable convexity condition we associate a knot concordance invariant ${\Upsilon }^C$. For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen’s $h_i$ invariants, and the Ozsváth–Stipsicz–Szabó upsilon invariant. Furthermore, to three such regions $C$, $C^{+}$ and $C^{-}$ we associate invariants ${\Upsilon }_{C^{\pm },C}$ generalizing the Kim–Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.