The braid index is not additive for the connected sum of 2-knots |
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| Authors: | Seiichi Kamada Shin Satoh Manabu Takabayashi |
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| Institution: | Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan ; Department of Mathematics, Chiba University, Inage, Chiba, 263-8522, Japan ; Japan Tokushima Prefectural, Mental Health & Welfare Center, 3-80 Shinkura, Tokushima, 770-0855, Japan |
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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  of  -knots  and  , it is easily seen that  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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| Keywords: | Braid index $2$-knots surface-knots $2$-dimensional braids braided surfaces charts |
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