Concordance invariant $\Upsilon$ for 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.