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Transactions of the American Mathematical Society

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

 

 

The braid index is not additive for the connected sum of 2-knots


Authors: Seiichi Kamada, Shin Satoh and Manabu Takabayashi
Journal: Trans. Amer. Math. Soc. 358 (2006), 5425-5439
MSC (2000): Primary 57Q45
DOI: https://doi.org/10.1090/S0002-9947-06-03867-0
Published electronically: April 11, 2006
MathSciNet review: 2238921
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Abstract: Any $ 2$-dimensional knot $ K$ can be presented in a braid form, and its braid index, $ {Braid}(K)$, is defined. For the connected sum $ K_1\char93 K_2$ of $ 2$-knots $ K_1$ and $ K_2$, it is easily seen that $ {Braid}(K_1\char93 K_2)\leq {B}(K_1) + {B}(K_2) -1$ holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of $ 1$-dimensional knots; the equality holds for $ 1$-knots. We prove that the equality does not hold for $ 2$-knots unless $ K_1$ or $ K_2$ is a trivial $ 2$-knot. We also prove that the $ 2$-knot obtained from a granny knot by Artin's spinning is of braid index $ 4$, and there are infinitely many $ 2$-knots of braid index $ 4$.


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Additional Information

Seiichi Kamada
Affiliation: Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan
Email: kamada@math.sci.hiroshima-u.ac.jp

Shin Satoh
Affiliation: Department of Mathematics, Chiba University, Inage, Chiba, 263-8522, Japan
Email: satoh@math.s.chiba-u.ac.jp

Manabu Takabayashi
Affiliation: Japan Tokushima Prefectural, Mental Health & Welfare Center, 3-80 Shinkura, Tokushima, 770-0855, Japan
Email: manabu12@khaki.plala.or.jp

DOI: https://doi.org/10.1090/S0002-9947-06-03867-0
Keywords: Braid index, $2$-knots, surface-knots, $2$-dimensional braids, braided surfaces, charts
Received by editor(s): July 15, 2003
Received by editor(s) in revised form: October 1, 2004
Published electronically: April 11, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.