Concordance invariant $\Upsilon$ for balanced spatial graphs using grid homology

TL;DR

This paper extends the combinatorial $$ invariant, originally defined for knots, to balanced spatial graphs using grid homology, providing bounds under cobordisms and confirming its concordance invariance.

Contribution

It introduces a new combinatorial $$ invariant for balanced spatial graphs and establishes its properties and bounds, expanding the scope of knot invariants.

Findings

The combinatorial $$ invariant is a concordance invariant for knots.

Bounds for $$ are provided when links are connected by cobordisms.

The invariant is extended to balanced spatial graphs using grid homology.

Abstract

The $\Upsilon$ invariant is a concordance invariant defined by using knot Floer homology. F\"{o}ldv\'{a}ri gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\Upsilon$ invariant for balanced spatial graph using grid homology for balanced spatial graph. Regarding links as spatial graphs, we give a upper and lower bounds for $\Upsilon$ when two links are connected by a cobordism. Also we show that the combinatorial $\Upsilon$ is a concordance invariant for knots.