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

### Citations:

Zbl 1146.57018; Zbl 1124.57003
Full Text:

### References:

 [1] 10.1007/s00029-017-0336-4 · Zbl 1456.57013 [2] 10.4310/MRL.2010.v17.n2.a2 · Zbl 1220.14008 [3] 10.1007/978-0-8176-4745-2_1 · Zbl 1198.53090 [4] 10.1007/s00208-015-1196-x · Zbl 1356.57013 [5] 10.1215/00127094-2008-012 · Zbl 1145.14016 [6] 10.1112/S0010437X1100724X · Zbl 1249.14005 [7] 10.1007/s00208-013-0984-4 · Zbl 1387.17027 [8] 10.1007/s00029-014-0162-x · Zbl 1370.17017 [9] ; Dunfield, Experiment. Math., 15, 129, (2006) [10] 10.2140/gtm.2012.18.309 · Zbl 1296.57014 [11] 10.1142/S0129167X07004400 · Zbl 1124.57003 [12] 10.4171/QT/1 · Zbl 1206.17015 [13] 10.2140/gt.2008.12.1387 · Zbl 1146.57018 [14] 10.4171/QT/84 · Zbl 1376.57015 [15] 10.1515/crelle-2012-0086 · Zbl 1352.16013 [16] ; Lusztig, Introduction to quantum groups. Progress in Mathematics, 110, (1993) · Zbl 0788.17010 [17] 10.1090/S0002-9947-2010-05155-4 · Zbl 1242.18016 [18] 10.2140/gt.2015.19.3031 · Zbl 1419.57027 [19] 10.4064/fm225-1-14 · Zbl 1336.57025 [20] 10.2140/gt.2017.21.2557 · Zbl 1427.17016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.