A diagrammatic grading bound for colored $\mathfrak{gl}_N$ link homology

Abstract: We establish a grading bound for colored $\mathfrak{gl}_N$ link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston--Bennequin number of a Legendrian link from $\mathfrak{gl}_N$ link homology, in the spirit of Ng.