Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Ribbon-moves for 2-knots with 1-handles attached and Khovanov-Jacobsson numbers


Authors: J. Scott Carter, Masahico Saito and Shin Satoh
Journal: Proc. Amer. Math. Soc. 134 (2006), 2779-2783
MSC (2000): Primary 57Q45; Secondary 57Q35
DOI: https://doi.org/10.1090/S0002-9939-06-08288-8
Published electronically: April 10, 2006
MathSciNet review: 2213759
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a crossing change along a double point circle on a $ 2$-knot is realized by ribbon-moves for a knotted torus obtained from the $ 2$-knot by attaching a $ 1$-handle. It follows that any $ 2$-knots for which the crossing change is an unknotting operation, such as ribbon $ 2$-knots and twist-spun knots, have trivial Khovanov-Jacobsson number.


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

  • 1. S. Asami and S. Satoh, An infinite family of non-invertible surfaces in $ 4$-space, Bull. London Math. Soc. 37 (2005), 285-296. MR 2119028
  • 2. D. Bar-Natan, Khovanov's homology for tangles and cobordisms, preprint available at: http://arxiv.org/pdf/math.GT/0410495
  • 3. J. Boyle, The turned torus knot in $ S^4$, J. Knot Theory Ramifications 2 (1993), 239-249. MR 1238874 (94i:57037)
  • 4. J.S. Carter and M. Saito, Knotted surfaces and their diagrams, Mathematical Surveys and Monographs, vol. 55, American Mathematical Society, Providence, RI, 1998. MR 1487374 (98m:57027)
  • 5. M. Jacobsson, An invariant of link cobordisms from Khovanov's homology theory, Algebr. Geom. Topol. 4 (2004), 1211-1251 MR 2113903
  • 6. T. Kanenobu and A. Shima, Two filtrations of ribbon 2-knots, Topology Appl. 121 (2002), 143-168. MR 1903688 (2003h:57034)
  • 7. A. Kawauchi, On pseudo-ribbon surface-links, J. Knot Theory Ramifications 11 (2002), 1043-1062. MR 1941684 (2003h:57035)
  • 8. M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (1999), 359-426.MR 1740682 (2002j:57025)
  • 9. -, An invariant of tangle cobordisms, preprint available at: http://xxx.lanl.gov/ abs/math.GT/0207264
  • 10. E. Ogasa, Ribbon-moves of $ 2$-knots: the Farber-Levine pairing and the Atiyah-Patodi-Sinder-Casson-Gordon-Ruberman $ \widetilde{\eta}$-invariants of $ 2$-knots, preprint available at: http://xxx.lanl.gov/abs/math.GT/0004007
  • 11. S. Satoh, Surface diagrams of twist-spun 2-knots, J. Knot Theory Ramifications 11 (2002), 413-430. MR 1905695 (2003e:57041)
  • 12. -, A note on unknotting numbers of twist-spun knots, Kobe J. Math. 21 (2004), 71-82.MR 2140603
  • 13. A. Shima, On simply knotted tori in $ S^4$ II, Knots '96 (Tokyo), 551-568, World Sci. Publishing, River Edge, NJ, 1997. MR 1664987 (99m:57022)
  • 14. M. Teragaito, Symmetry-spun tori in the four-sphere, Knots 90 (Osaka, 1990), 163-171, de Gruyter, Berlin, 1992. MR 1177421 (93g:57029)
  • 15. T. Yajima, On simply knotted spheres in $ R^4$, Osaka J. Math. 1 (1964), 133-152.MR 0172280 (30:2500)
  • 16. T. Yashiro, Deformations of surface diagrams, talk at First KOOK Seminar International Knot Theory and Related Topics, July 2004.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57Q45, 57Q35

Retrieve articles in all journals with MSC (2000): 57Q45, 57Q35


Additional Information

J. Scott Carter
Affiliation: Department of Mathematics, University of South Alabama, Mobile, Alabama 36688
Email: carter@jaguar1.usouthal.edu

Masahico Saito
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: saito@math.usf.edu

Shin Satoh
Affiliation: Graduate School of Science and Technology, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan
Email: satoh@math.s.chiba-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-06-08288-8
Keywords: Khovanov homology, 2-knot, ribbon-move, twist-spun knot, crossing change.
Received by editor(s): October 19, 2004
Received by editor(s) in revised form: April 14, 2005
Published electronically: April 10, 2006
Additional Notes: The first author was supported in part by NSF Grant DMS $#0301095$.
The second author was supported in part by NSF Grant DMS $#0301089$.
The third author was supported in part by JSPS Postdoctoral Fellowships for Research Abroad.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society