Honam Mathematical Journal (호남수학학술지)

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RASMUSSEN INVARIANTS OF SOME 4-STRAND PRETZEL KNOTS

  • KIM, SE-GOO (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University) ;
  • YEON, MI JEONG (Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University)
  • Received : 2015.02.27
  • Accepted : 2015.04.03
  • Published : 2015.06.25
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Abstract

It is known that there is an infinite family of general pretzel knots, each of which has Rasmussen s-invariant equal to the negative value of its signature invariant. For an instance, homologically ${\sigma}$-thin knots have this property. In contrast, we find an infinite family of 4-strand pretzel knots whose Rasmussen invariants are not equal to the negative values of signature invariants.

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References

  1. T. Abe and K. Kishimoto, The dealternating number and the alternation number of a closed 3-braid, J. Knot Theory Ramifications 19 (2010), no. 9, 1157-1181. https://doi.org/10.1142/S0218216510008352
  2. M. M. Asaeda and J. H. Przytycki, Khovanov homology: torsion and thick- ness, Advances in topological quantum field theory, 135-166, NATO Sci. Ser. II Math. Phys. Chem., 179, Kluwer Acad. Publ., Dordrecht, 2004.
  3. A. Beliakova and S. Wehrli, Categorification of the colored Jones polynomial and Rasmussen invariant of links, Canad. J. Math. 60 (2008), no. 6, 1240-1266. https://doi.org/10.4153/CJM-2008-053-1
  4. J. Greene, Homologically thin, non-quasi-alternating links, Math. Res. Lett. 17 (2010), no. 1, 39-49. https://doi.org/10.4310/MRL.2010.v17.n1.a4
  5. S. Jabuka, Rational Witt classes of pretzel knots, Osaka J. Math. 47 (2010), no. 4, 977-1027.
  6. T. Kawamura, An estimate of the Rasmussen invariant for links and the deter mination for certain links, to appear in Topology Appl.
  7. M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359-426. https://doi.org/10.1215/S0012-7094-00-10131-7
  8. S.-G. Kim and M. J. Yeon, Rasmussen and Ozsvath-Szabo invariants of a family of general pretzel knots, to appear in J. Knot Theory Ramifications.
  9. The Knot Atlas, http://katlas.org/
  10. E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 554-586. https://doi.org/10.1016/j.aim.2004.10.015
  11. A. Manion, The rational Khovanov homology of 3-strand pretzel links, J. Knot Theory Ramifications 23 (2014), no. 8, 1450040, 40 pp.
  12. C. Manolescu and P. Ozsvath, On the Khovanov and knot Floer homologies of quasi-alternating links, Proceedings of Gokova Geometry-Topology Conference 2007, 60-81, Gokova Geometry/Topology Conference (GGT), Gokova, 2008.
  13. J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419-447 https://doi.org/10.1007/s00222-010-0275-6
  14. R. Suzuki, Khovanov homology and Rasmussen's s-invariants for pretzel knots. J. Knot Theory Ramifications 19 (2010), no. 9, 1183-1204. https://doi.org/10.1142/S0218216510008376
  15. P. Turner, Five lectures on Khovanov homology, preprint (arXiv:math/0606464).
  16. O. Viro, Remarks on definition of Khovanov homology, preprint (arXiv:math/0202199).