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Remarks on coloured triply graded link invariants. (English) Zbl 1381.57010

M. Khovanov and L. Rozansky [Geom. Topol. 12, No. 3, 1387–1425 (2008; Zbl 1146.57018)] and M. Khovanov [Int. J. Math. 18, No. 8, 869–885 (2007; Zbl 1124.57003)] constructed a triply-graded homology theory of links which categorifies the HOMFLYPT polynomial. The paper under review demonstates how to combine categorical \(\mathfrak{sl}_N\) actions, cabling and infinite twists to construct a triply-graded link invariant categorifying the HOMFLYPT polynomial of a link whose components are colored with arbitrary partitions. Moreover, the author constructs additional differentials on the triply-graded invariant and proves that the associated spectral sequence converges to a doubly-graded link invariant. The author claims that the resulting doubly-graded invariant categorifies the Reshetikhin-Turaev link invariant associated to appropriate symmetric powers of the fundamental representation of \(U_q(\mathfrak{sl}_N)\).

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
16T99 Hopf algebras, quantum groups and related topics
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