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ISSN 1088-6850(e) ISSN 0002-9947(p)

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
Posted: April 11, 2006
MathSciNet review: 2238921
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Abstract: Any -dimensional knot can be presented in a braid form, and its braid index, , is defined. For the connected sum $K_1\char93 K_2$ of -knots and , 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 -dimensional knots; the equality holds for -knots. We prove that the equality does not hold for -knots unless or is a trivial -knot. We also prove that the -knot obtained from a granny knot by Artin's spinning is of braid index , and there are infinitely many -knots of braid index .

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