Remarks on coloured triply graded link invariants

Abstract

We explain how existing results (such as categorical ${\mathfrak{s}\mathfrak{l}}_n$ actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the homfly polynomial of links coloured by arbitrary partitions. The construction uses a categorified homfly clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures.