Jones’ braid-plat formula and a new surgery triple

Birman, Joan S.; Kanenobu, Taizo

doi:10.1090/S0002-9939-1988-0929004-1

Jones’ braid-plat formula and a new surgery triple
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by Joan S. Birman and Taizo Kanenobu PDF

Proc. Amer. Math. Soc. 102 (1988), 687-695 Request permission

Abstract:

A link ${L_\beta }\left ( {2k,n - 2k} \right )$ is defined by a type $\left ( {2k,n - 2k} \right )$ pairing of an $n$-braid $\beta$ if the first $2k$ strands are joined up as in a plat and the remaining $n - 2k$ as in a closed braid. The main result is a formula for the Jones polynomials of ${L_\beta }\left ( {2k,n - 2k} \right )$, valid for all $k,0 \leqslant 2k \leqslant n$, which generalizes and relates earlier results of Jones for the cases $n = 0$ and $2k$.

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