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

Kamada, Seiichi; Satoh, Shin; Takabayashi, Manabu

doi:10.1090/S0002-9947-06-03867-0

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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