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On sutured Khovanov homology and axis-preserving mutations

Diana Hubbard

Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA

https://doi.org/10.1142/S0218216517500171Cited by:3 (Source: Crossref)

Abstract

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures.

Keywords:

AMSC: 57M27, 20F36

References

1. M. Asaeda, J. Przytycki and A. Sikora, Categorification of the Kauffman bracket skein module of I-bundles over surfaces, Algebra Geom. Topol. 4(52) (2004) 1177–1210. Crossref, Google Scholar
2. J. Baldwin and J. Elisenda Grigsby, Categorified invariants and the braid group, Proc. Amer. Math. Soc. 143(7) (2015) 2801–2814. Crossref, Web of Science, Google Scholar
3. J. Birman and T. Brendle, Braids: a survey, Handbook of Knot Theory, eds. W. Menasco and M. Thistlethwaite (Elsevier B.V., Amsterdam, 2005), pp. 19–103. Crossref, Google Scholar
4. J. S. Birman and W. W. Menasco, Studying links via closed braids. V. the unlink, Trans. Amer. Math. Soc. 329(2) (1992) 585–606. Crossref, Web of Science, Google Scholar
5. J. S. Birman and W. W. Menasco, A note on closed 3-braids, Commun. Contemp. Math. 10(supp01) (2008) 1033–1047. Link, Google Scholar
6. J. Bloom, Odd Khovanov homology is mutation invariant, Math. Res. Letters, 17(1) 1–10. Crossref, Web of Science, Google Scholar
7. J. E. Grigsby and Yi. Ni, Sutured Khovanov homology distinguishes braids from other tangles, Math. Res. Lett. 21(6) (2014) 1263–1275. Crossref, Web of Science, Google Scholar
8. J. E. Grigsby and S. Wehrli, On gradings in Khovanov homology and sutured Floer homology, Topology and Geometry in Dimension Three, eds. W. Li, L. Bartolini, J. Johnson, F. Luo, R. Myers and J. H. Rubenstein, Vol. 560 (American Mathematical Society, Providence, RI. Series: Comtemporary Mathematics, 2011), pp. 111–128. Crossref, Google Scholar
9. C. Jackson and T. Kerler, The Lawrence–Krammer–Bigelow representations of the braid groups via Uq (sl2), Adv. Math. 228(3) (2011) 1689–1717. Crossref, Web of Science, Google Scholar
10. C. H. Jackson, Braid group representations, Ph.D. thesis, The Ohio State University (2001). Google Scholar
11. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126(2) (1987) 335–388. Crossref, Web of Science, Google Scholar
12. T. Kanenobu, The homfly and the Kauffman bracket polynomials for the generalized mutant of a link, Topol. Appl. 61(3) (1995) 257–279. Crossref, Web of Science, Google Scholar
13. T. Khandhawit and L. Ng, A family of transversely nonsimple knots, Algebraic Geom. Topol. 10(1) (2010) 293–314. Crossref, Web of Science, Google Scholar
14. M. Khovanov, Graphical calculus, canonical bases and Kazhdan-Lusztig theory, Ph.D. thesis, Yale University, (1997). Google Scholar
15. M. Khovanov and P. Seidel, Quivers, floer cohomology, and braid group actions, J. Amer. Math. Soc. 15(1) (2002) 203–271. Crossref, Web of Science, Google Scholar
16. L. Ng, P. Ozsváth and D. Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6(4) (2008) 461–490. Crossref, Web of Science, Google Scholar
17. L. Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17(1) (2013) 413–467. Crossref, Web of Science, Google Scholar
18. S. Wehrli, Mutation invariance of Khovanov homology over $f_{2}$ $f_{2}$ , Quantum Topol. 1(2) (2010) 111–128. Crossref, Google Scholar