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 -knot is realized by ribbon-moves for a knotted torus obtained from the -knot by attaching a -handle. It follows that any -knots for which the crossing change is an unknotting operation, such as ribbon -knots and twist-spun knots, have trivial Khovanov-Jacobsson number.

**1.**S. Asami and S. Satoh,*An infinite family of non-invertible surfaces in -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*, 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 -knots: the Farber-Levine pairing and the Atiyah-Patodi-Sinder-Casson-Gordon-Ruberman -invariants of -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 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*, 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.

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