The lower bound of the -indices of surface links via quandle cocycle invariants

Author:
Masahide Iwakiri

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1189-1210

MSC (2000):
Primary 57Q45

DOI:
https://doi.org/10.1090/S0002-9947-09-04769-2

Published electronically:
September 23, 2009

MathSciNet review:
2563726

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Abstract: The -index of a surface link is the minimal number of the triple points of surface braids representing . In this paper, for a given -cocycle, we consider the minimal number of the -indices of surface links whose quandle cocycle invariants associated with are non-trivial, and denote it . In particular, we show that and , where is Mochizuki's -cocycle of the dihedral quandle of order and is an odd prime integer . As a consequence, for a given non-negative integer , there are surface knots with genus with the -index .

**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 (2005k:57044)****2.**J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito,*Quandle cohomology and state-sum invariants of knotted curves and surfaces*, Trans. Amer. Math. Soc.,**355**(2003), 3947-3989. MR**1990571 (2005b:57048)****3.**J. S. Carter, D. Jelsovsky, S. Kamada and M. Saito,*Computations of quandle cocycle invariants of knotted curves and surfaces*, Adv. in Math.,**157**(2001), 36-94. MR**1808844 (2001m:57009)****4.**J. S. Carter, M. Saito and S. Satoh,*Ribbon concordance of surface-knots via quandle cocycle invariants*, J. Aust. Math. Soc.**80**(2006), 131-147. MR**2212320 (2006k:57066)****5.**R. Fenn, C. Rourke and B. Sanderson,*James bundles and applications*, Proc. London Math. Soc. (3)**89**(2004), 217-240. MR**2063665 (2005d:55006)****6.**I. Hasegawa,*The lower bound of the -indices of non-ribbon surface-links*, Osaka J. Math.,**41**(2004), 891-909. MR**2116344 (2005k:57045)****7.**E. Hatakenaka,*An estimate of the triple point numbers of surface-knots by quandle cocycle invariants*, Topology Appl.**139**(2004), 129-144. MR**2051101 (2005d:57036)****8.**S. Kamada,*Surfaces in of braid index three are ribbon*, J. Knot Theory Ramifications,**1**(1992), 137-160. MR**1164113 (93h:57039)****9.**S. Kamada,*-dimensional braids and chart descriptions*, Topics in Knot Theory, 277-287, NATO ASI series C,**399**(Erzurum/Turkey 1992), Kluwer Academic Publishers, 1992. MR**1257915****10.**S. Kamada,*A characterization of groups of closed orientable surfaces in -space*, Topology,**33**(1994), 113-122. MR**1259518 (95a:57002)****11.**S. Kamada,*An observation of surface braids via chart description*, J. Knot Theory Ramifications,**4**(1996), 517-529. MR**1406718 (97j:57009)****12.**S. Kamada, Braid and knot theory in dimension four, Math. Surveys Monogr.**95**, Amer. Math. Soc., 2002. MR**1900979 (2003d:57050)****13.**T. Mochizuki,*Some calculations of cohomology groups of finite Alexander quandles*, J. Pure Appl. Algebra**179**(2003), 287-330. MR**1960136 (2004b:55013)****14.**T. Nagase, A. Shima,*Properties of minimal charts and their applications. I*, J. Math. Sci. Univ. Tokyo**14**(2007), 69-97. MR**2320385 (2008c:57040)****15.**M. Ochiai, T. Nagase, A. Shima,*There exists no minimal -chart with five white vertices*, Proc. Sch. Sci. Tokai Univ.**40**(2005), 1-18. MR**2138333 (2006b:57035)****16.**S. Satoh, A. Shima,*The -twist-spun trefoil has the triple point number four*, Trans. Amer. Math. Soc.**356**(2004), 1007-1024. MR**1984465 (2004k:57032)****17.**S. Satoh, A. Shima,*Triple point numbers and quandle cocycle invariants of knotted surfaces in -space*, New Zealand J. Math.**34**(2005), 71-79. MR**2141479 (2006e:57031)****18.**O. Ya. Viro, Lecture given at Osaka City University, September, 1990.

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Additional Information

**Masahide Iwakiri**

Affiliation:
Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka 558-8585, Japan

Email:
iwakiri@sci.osaka-cu.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-09-04769-2

Received by editor(s):
June 4, 2007

Published electronically:
September 23, 2009

Additional Notes:
This paper was supported by JSPS Research Fellowships for Young Scientists and the 21 COE program “Constitution of wide-angle mathematical basis focused on knots”.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.