Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Braids, transversal links and the Khovanov-Rozansky Theory


Author: Hao Wu
Journal: Trans. Amer. Math. Soc. 360 (2008), 3365-3389
MSC (2000): Primary 57M25, 57R17
DOI: https://doi.org/10.1090/S0002-9947-08-04339-0
Published electronically: February 27, 2008
MathSciNet review: 2386230
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish some inequalities for the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact $ S^3$ which are sharper than the well-known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants and discuss some of their properties.


References [Enhancements On Off] (What's this?)

  • 1. D. Bar-Natan, Khovanov's homology for tangles and cobordisms, Algebr. Geom. Topol. 9 (2005), 1443-1499 (electronic). MR 2174270 (2006g:57017)
  • 2. D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-108 (1983), 87-161. MR 753131 (86e:58070)
  • 3. E. Ferrand, On Legendrian knots and polynomial invariants, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1169-1176 (electronic). MR 1873793 (2002j:57047)
  • 4. J. Franks, R. F. Williams, Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), no. 1, 97-108. MR 896009 (88k:57006)
  • 5. D. Fuchs, S. Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), no. 5, 1025-1053. MR 1445553 (99a:57006)
  • 6. M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. MR 1740682 (2002j:57025)
  • 7. M. Khovanov, L. Rozansky, Matrix factorizations and link homology, arXiv:math.QA/ 0401268.
  • 8. M. Khovanov, L. Rozansky, Matrix factorizations and link homology II, arXiv:math.QA/ 0505056.
  • 9. H. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107-109. MR 809504 (87c:57006)
  • 10. S. Orevkov, V. Shevchishin, Markov theorem for transversal links, J. Knot Theory Ramifications 12 (2003), no. 7, 905-913. MR 2017961 (2004j:57011)
  • 11. O. Plamenevskaya, Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006), 571-586. MR 2250492 (2007d:57043)
  • 12. N. Wrinkle, The Markov Theorem for transverse knots, arXiv:math.GT/0202055.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M25, 57R17

Retrieve articles in all journals with MSC (2000): 57M25, 57R17


Additional Information

Hao Wu
Affiliation: Department of Mathematics and Statistics, Lederle Graduate Research Tower, 710 North Pleasant Street, University of Massachusetts, Amherst, Massachusetts 01003-9305
Address at time of publication: Department of Mathematics, The George Washington University, Monroe Hall, Room 240, 2115 G Street, N.W., Washington, DC 20052
Email: wu@math.umass.edu, haowu@gwu.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04339-0
Keywords: Braid, transversal knot, knot homology, matrix factorization
Received by editor(s): January 20, 2006
Received by editor(s) in revised form: May 24, 2006
Published electronically: February 27, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society