Upsilon invariant for graphs and the homology cobordism group of homology cylinders

TL;DR

This paper generalizes the Upsilon invariant from knots to embedded graphs in rational homology spheres, using tangle Floer homology to define a new homomorphism on the homology cobordism group of homology cylinders.

Contribution

It introduces a novel invariant extending Upsilon to graphs, establishing its homomorphism property and applications in the context of homology cylinders.

Findings

Invariant induces a homomorphism on the homology cobordism group

Lifts relative gradings to absolute gradings for certain tangles

Provides a concatenation formula for the invariant

Abstract

Upsilon is a homomorphism on the smooth concordance group of knots defined by Ozsv\'{a}th, Stipsicz and Szab\'{o}. In this paper, we define a generalization of upsilon for a family of embedded graphs in rational homolog spheres. We show that our invariant will induce a homomorphism on the homology cobordism group of homology cylinders, and present some applications. To define this invariant, we use tangle Floer homology. We lift relative gradings on tangle Floer homology to absolute gradings (for certain tangles) and prove a concatenation formula for it.