## 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 $4$.## References

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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