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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The lower bound of the $ \lowercase{w}$-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 $ w$-index of a surface link $ F$ is the minimal number of the triple points of surface braids representing $ F$. In this paper, for a given $ 3$-cocycle, we consider the minimal number of the $ w$-indices of surface links whose quandle cocycle invariants associated with $ f$ are non-trivial, and denote it $ \omega(f)$. In particular, we show that $ \omega(\theta_3)=6$ and $ \omega(\theta_p)\geq 7$, where $ \theta_n$ is Mochizuki's $ 3$-cocycle of the dihedral quandle of order $ n$ and $ p$ is an odd prime integer $ \not=3$. As a consequence, for a given non-negative integer $ g$, there are surface knots with genus $ g$ with the $ w$-index $ 6$.


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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.