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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The braid index is not additive for the connected sum of 2-knots
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by Seiichi Kamada, Shin Satoh and Manabu Takabayashi PDF
Trans. Amer. Math. Soc. 358 (2006), 5425-5439 Request permission

Abstract:

Any $2$-dimensional knot $K$ can be presented in a braid form, and its braid index, $\operatorname {Braid}(K)$, is defined. For the connected sum $K_1\# K_2$ of $2$-knots $K_1$ and $K_2$, it is easily seen that $\operatorname {Braid}(K_1\# K_2)\leq \operatorname {B}(K_1) + \operatorname {B}(K_2) -1$ 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
  • MR Author ID: 288529
  • 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
  • Received by editor(s): July 15, 2003
  • Received by editor(s) in revised form: October 1, 2004
  • Published electronically: April 11, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 5425-5439
  • MSC (2000): Primary 57Q45
  • DOI: https://doi.org/10.1090/S0002-9947-06-03867-0
  • MathSciNet review: 2238921