No Access

Categorification of the colored ${𝔰 𝔩}_{3}$ -invariant

Louis-Hadrien Robert

Universität Hamburg Bundesstrasse 55, 20146 Hamburg, Germany

E-mail Address: louis-hadrien.robert@uni-hamburg.de

https://doi.org/10.1142/S0218216516500383Cited by:2 (Source: Crossref)

Abstract

We give explicit resolutions of all finite-dimensional simple $U_{q} ({𝔰 𝔩}_{3})$ -modules. We use these resolutions to categorify the colored ${𝔰 𝔩}_{3}$ -invariant of framed links via a complex of complexes of graded $ℤ$ -modules.

Keywords:

AMSC: 57M25, 22E46, 20G42, 05E10

References

1. D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337–370 (electronic). Crossref, Google Scholar
2. D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443–1499 (electronic). Crossref, Web of Science, Google Scholar
3. A. Beliakova and S. Wehrli, Categorification of the colored Jones polynomial and Rasmussen invariant of links, Canad. J. Math. 60(6) (2008) 1240–1266. Crossref, Web of Science, Google Scholar
4. C. Caprau, A cohomology theory for colored tangles, in Knots in Poland. III, Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science, Warsaw, 2014), pp. 13–25. Crossref, Google Scholar
5. J. S. Carter, Reidemeister/Roseman-type moves to embedded foams in 4-dimensional space, preprint (2012), arXiv:1210.3608. Google Scholar
6. J. S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(3) (2002) 311–322. Link, Web of Science, Google Scholar
7. D. Clark, Functoriality for the ${𝔰 𝔲}_{3}$ Khovanov homology, Algebr. Geom. Topol. 9(2) (2009) 625–690. Crossref, Web of Science, Google Scholar
8. C. Kassel, M. Rosso and V. Turaev, Quantum Groups and Knot Invariants, Panoramas et Synthése, No. 5 (Société Mathématique de France, 1997). Google Scholar
9. M. Khovanov, ${𝔰 𝔲}_{3}$ link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081. Crossref, Google Scholar
10. M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14(1) (2005) 111–130. Link, Web of Science, Google Scholar
11. G. Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180(1) (1996) 109–151. Crossref, Web of Science, Google Scholar
12. L. Lewark, ${𝔰 𝔩}_{3}$ -foam homology calculations, Algebr. Geom. Topol. 13(6) (2013) 3661–3686. Crossref, Web of Science, Google Scholar
13. L. Lewark, Les homologies de Khovanov–Rozansky, toiles nouées pondérées et le genre lisse, Ph.D. thesis, Univeristé Paris 7 — Denis Diderot (2013). Google Scholar
14. M. Mackaay and P. Turner, Bar-Natan’s Khovanov homology for coloured links, Pacific J. Math. 229(2) (2007) 429–446. Crossref, Web of Science, Google Scholar
15. M. Mackaay and P. Vaz, The universal ${sl}_{3}$ -link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169. Crossref, Web of Science, Google Scholar
16. M. Mackaay and P. Vaz, The foam and the matrix factorization ${sl}_{3}$ link homologies are equivalent, Algebr. Geom. Topol. 8(1) (2008) 309–342. Crossref, Web of Science, Google Scholar
17. S. Morrison and A. Nieh, On Khovanov’s cobordism theory for ${𝔰 𝔲}_{3}$ knot homology, J. Knot Theory Ramifications 17(9) (2008) 1121–1173. Link, Web of Science, Google Scholar
18. T. Ohtsuki, Quantum Invariants, Series on knots and everything, No. 29 (World Scientific, 2002). Google Scholar
19. P. Ozsvath, J. Rasmussen and Z. Szabo, Odd khovanov homology, preprint (2007), arXiv:0710.4300. Google Scholar
20. H. Queffelec and D. E. V. Rose, The ${𝔰 𝔩}_{n}$ foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality, preprint (2014), arXiv:1405.5920. Google Scholar
21. N. Y. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26. Crossref, Web of Science, Google Scholar
22. L.-H. Robert, Sur l’homologie ${𝔰 𝔩}_{3}$ des enchevetrements; algébre de Khovanov–Kuperberg, Ph.D. thesis, Université Paris 7 — Denis Diderot (2013). Google Scholar
23. L.-H. Robert, A characterization of indecomposable web modules over Khovanov–Kuperberg algebras, Algebr. Geom. Topol. 15(3) (2015) 1303–1362. Crossref, Web of Science, Google Scholar
24. D. E. V. Rose, Categorification of quantum sl3 projectors and the sl3 Reshetikhin–Turaev invariant of framed tangles, Ph.D. thesis, (2012) Duke University. Google Scholar