On a triply graded Khovanov homology (Q2799085)

Given a link in the 3-sphere, \textit{M. Khovanov} [Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] showed how to associate to it a bigraded abelian group whose isomorphism class is an isotopy invariant of the link. The bigraded abelian group is called the Khovanov homology of the link, and it is a `categorification' of the Jones polynomial in the sense that the graded Euler characteristic of Khovanov homology agrees with this polynomial. An alternative categorification, called odd Khovanov homology, was produced by \textit{P. S. Ozsváth} et al. [Algebr. Geom. Topol. 13, No. 3, 1465--1488 (2013; Zbl 1297.57032)] using a different algebraic construction. In a previous paper [Banach Cent. Publ. 103, 291--355 (2014; Zbl 1336.57024)], the author constructed a tangle cobordism category, in the spirit of \textit{D. Bar-Natan} [Algebr. Geom. Topol. 2, 337--370 (2002; Zbl 0998.57016)]. His construction produces for each link a bigraded group which he calls \textit{generalised Khovanov homology}, and which can be shown to specialise to each of classical and odd Khovanov homology, thereby generalising them into one setting.NEWLINENEWLINEIn the paper under review, the author observes that one can triply grade the generalised Khovanov homology by splitting the non-homological grading on the chain level into two gradings, which are themselves defined using the underlying TQFT. The triple grading turns out to give no more information than the bigraded version when you pass to homology of a link. However, the author shows the triply graded chain complex can be used to make more sensitive analysis of algebraic operations one might want to perform, corresponding to operations on link diagrams. In particular, the author produces formulae for the generalised Khovanov homology of disjoint unions and connected sums of links, in terms of certain derived tensor products of the chain complexes of the link summands. These formulae then specialise to formulae for each of classical and odd Khovanov homology -- in the latter case this is a new result. The author then goes on to analyse the effect, in the generalised setting and with integral coefficients, of sliding basepoints over crossings in the link diagram.