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

Published electronically:
April 10, 2006

MathSciNet review:
2213759

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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.**Soichiro Asami and Shin Satoh,*An infinite family of non-invertible surfaces in 4-space*, Bull. London Math. Soc.**37**(2005), no. 2, 285–296. MR**2119028**, 10.1112/S0024609304003832**2.**D. Bar-Natan,*Khovanov's homology for tangles and cobordisms,*preprint available at:`http://arxiv.org/pdf/math.GT/0410495`**3.**Jeffrey Boyle,*The turned torus knot in 𝑆⁴*, J. Knot Theory Ramifications**2**(1993), no. 3, 239–249. MR**1238874**, 10.1142/S0218216593000155**4.**J. Scott Carter and Masahico Saito,*Knotted surfaces and their diagrams*, Mathematical Surveys and Monographs, vol. 55, American Mathematical Society, Providence, RI, 1998. MR**1487374****5.**Magnus Jacobsson,*An invariant of link cobordisms from Khovanov homology*, Algebr. Geom. Topol.**4**(2004), 1211–1251 (electronic). MR**2113903**, 10.2140/agt.2004.4.1211**6.**Taizo Kanenobu and Akiko Shima,*Two filtrations of ribbon 2-knots*, Proceedings of the First Joint Japan-Mexico Meeting in Topology (Morelia, 1999), 2002, pp. 143–168. MR**1903688**, 10.1016/S0166-8641(01)00115-8**7.**Akio Kawauchi,*On pseudo-ribbon surface-links*, J. Knot Theory Ramifications**11**(2002), no. 7, 1043–1062. MR**1941684**, 10.1142/S0218216502002128**8.**Mikhail Khovanov,*A categorification of the Jones polynomial*, Duke Math. J.**101**(2000), no. 3, 359–426. MR**1740682**, 10.1215/S0012-7094-00-10131-7**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.**Shin Satoh,*Surface diagrams of twist-spun 2-knots*, J. Knot Theory Ramifications**11**(2002), no. 3, 413–430. Knots 2000 Korea, Vol. 1 (Yongpyong). MR**1905695**, 10.1142/S0218216502001718**12.**Shin Satoh,*A note on unknotting numbers of twist-spun knots*, Kobe J. Math.**21**(2004), no. 1-2, 71–82. MR**2140603****13.**Akiko Shima,*On simply knotted tori in 𝑆⁴. II*, KNOTS ’96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 551–568. MR**1664987****14.**Masakazu Teragaito,*Symmetry-spun tori in the four-sphere*, Knots 90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 163–171. MR**1177421****15.**Takeshi Yajima,*On simply knotted spheres in 𝑅⁴*, Osaka J. Math.**1**(1964), 133–152. MR**0172280****16.**T. Yashiro,*Deformations of surface diagrams*, talk at First KOOK Seminar International Knot Theory and Related Topics, July 2004.

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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:
http://dx.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