On the Upsilon invariant in grid homology

TL;DR

This paper proves the equivalence of the Upsilon invariant in knot Floer homology and grid homology, and explores its properties within the grid homology framework, enhancing understanding of knot invariants.

Contribution

It establishes the equivalence of Upsilon invariants across two homology theories and investigates their properties in grid homology, providing new insights.

Findings

Upsilon invariants in knot Floer and grid homology are equivalent.

Properties of the Upsilon invariant are characterized within grid homology.

Enhances understanding of knot concordance invariants.

Abstract

The Upsilon invariant is a concordance invariant in knot Floer homology. F\"{o}ldv\'{a}ri reconstructed the Upsilon invariant using grid homology. We prove that the Upsilon invariant in knot Floer homology and one in grid homology are equivalent. Furthermore, we show some properties of the Upsilon invariant in the framework of grid homology.