Meridian twisting of closed braids and the Homfly polynomial
Meridian twisting of closed braids and the Homfly polynomial
复制标题
闭合辫子的经络扭转和 Homfly 多项式
DOI：
10.1017/s0305004108002016
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发表时间：
2008
影响因子：
0.8
通讯作者：
T. Kalmán
中科院分区：
文献类型：
--
作者：
T. Kalmán
摘要 设 β 为 n 股辫子，指数和为 w。令 Δ 为 Garside 半捻辫子。我们证明β闭包的Homfly多项式中vw−n+1的系数与βΔ2闭包的Homfly多项式中vw+n2−1的系数的(−1)n−1倍一致。这种巧合意味着，当且仅当对于 $\widehat{\beta\Delta^2}$ 的 Homfly 多项式的 v 次数而言，较高的 MFW 估计是尖锐的时，$\widehat\beta$ 的 Homfly 多项式的 v 次数的较低 Morton-Franks-Williams 估计是尖锐的。
Abstract Let β be a braid on n strands, with exponent sum w. Let Δ be the Garside half-twist braid. We prove that the coefficient of vw−n+1 in the Homfly polynomial of the closure of β agrees with (−1)n−1 times the coefficient of vw+n2−1 in the Homfly polynomial of the closure of βΔ2. This coincidence implies that the lower Morton–Franks–Williams estimate for the v–degree of the Homfly polynomial of $\widehat\beta$ is sharp if and only if the upper MFW estimate is sharp for the v–degree of the Homfly polynomial of $\widehat{\beta\Delta^2}$.