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Khovanov-Rozansky homology and the braid index of a knot
Author(s):
Keiko
Kawamuro
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2459-2469.
MSC (2000):
Primary 57M25, 57M27;
Secondary 57M50
Posted:
February 23, 2009
MathSciNet review:
2495283
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Abstract:
We construct a knot whose braid index is not detected by the Morton-Franks-Williams (MFW) inequality but is detected by a related KR-MFW inequality that comes from the Khovanov-Rozansky homology. We also construct infinitely many knots whose braid indices are not detected by the KR-MFW inequality.
References:
-
- 1.
- Bennequin, D., Entrelacements et équations de Pfaff, Astérisque 107-108 (1983), 87-161. MR 753131 (86e:58070)
- 2.
- Birman, J. S. and Menasco, W. W., Studying links via closed braids, III: Classifying links which are closed
 -braids, Pacific J. Math. 161 (1993), no. 1, 25-113. MR 1237139 (94i:57005) - 3.
- Dunfield, N. M., Gukov, S. and Rasmussen, J., The superpolynomial for knot homologies, Experiment. Math. 15 (2006), no. 2, 129-159. MR 2253002 (2008c:57020)
- 4.
- Elrifai, E. A.,
Positive braids and Lorenz links, Ph.D. thesis, Liverpool University, 1988. - 5.
- Franks, J. and Williams, R. F., Braids and the Jones polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. MR 896009 (88k:57006)
- 6.
- Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335-388. MR 908150 (89c:46092)
- 7.
- Kálmán, T., Meridian twisting of closed braids and the HOMFLY polynomial, arXiv:0803.0103
- 8.
- Kawamuro, K., The algebraic crossing number and the braid index of knots and links, Algebraic & Geometric Topology 6 (2006), 2313-2350. MR 2286028 (2007k:57012)
- 9.
- Kawamuro, K., Conjectures on the braid index and the algebraic crossing number, Intelligence of Low Dimensional Topology 2006, Knots and Everything, 40, World Scientific Publ. Co., Hackensack, NJ, 2007, 151-155. MR 2371720
- 10.
- Khovanov, M. and Rozansky, L., Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1-91. MR 2391017
- 11.
- Khovanov, M. and Rozansky, L., Matrix factorizations and link homology. II, Geom. Topol. 12 (2008), no. 3, 1387-1425.
- 12.
- Malešič, J. and Traczyk, P., Seifert circles, braid index and the algebraic crossing number, Topology Appl. 153 (2005), no. 2-3, 303-317. MR 2175354 (2006i:57013)
- 13.
- Morton, H. R., Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), 107-109. MR 809504 (87c:57006)
- 14.
- Murasugi, K., On the braid index of alternating links, Trans. Amer. Math. Soc. 326 (1991), no. 1, 237-260. MR 1000333 (91j:57009)
- 15.
- Rasmussen, J., Some differentials on Khovanov-Rozansky homology, math.GT/0607544
- 16.
- Rudolph, L., Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51-59. MR 1193540 (94d:57028)
- 17.
- Shumakovitch, A., KhoHo: a program for computing Khovanov homology, www.geometrie.ch/ KhoHo/
- 18.
- Stoimenow, A., Properties of closed
 -braids, math/0606435 - 19.
- Wu, H., Braids, transversal links and the Khovanov-Rozansky theory, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3365-3389. MR 2386230
- 20.
- Yamada, S., The minimal number of Seifert circles equals the braid index of a link, Invent. Math. 89 (1987), no. 2, 347-356. MR 894383 (88f:57015)
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Additional Information:
Keiko
Kawamuro
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005
Address at time of publication:
School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey 08540
Email:
kk6@ias.edu
DOI:
10.1090/S0002-9939-09-09743-3
PII:
S 0002-9939(09)09743-3
Received by editor(s):
November 9, 2007,
Received by editor(s) in revised form:
July 2, 2008
Posted:
February 23, 2009
Additional Notes:
The author was partially supported by NSF grants DMS-0806492 and DMS-0635607.
Communicated by:
Daniel Ruberman
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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