AI 中文总结
本文证明双手性纽结的辫指标必为奇数，否定Stoimenow猜想，并完整分类辫指标3的素双手性纽结，给出基于标准3-辫词回文性或奇周期的对称性判据。
AI 中文摘要
我们研究纽结的双手性如何约束其辫指标，并在最小非平凡情形下约束其辫词组合。利用Dynnikov--Prasolov对Jones猜想的解决，我们证明双手性纽结的辫指标为奇数。这否定性地回答了Stoimenow关于偶数辫指标双手性纽结的问题。随后，我们对辫指标为3的素双手性纽结进行分类。每个辫指标为3的双手性纽结都是交错纽结，并允许一个由词c编码的标准形式的最小3-辫表示。基于Birman--Menasco对闭3-辫的分类和Murasugi正规形式，我们描述了c上的二面体作用，其中镜像和镜像反转操作分别由旋转和反射实现。对于闭包为辫指标3的素纽结的标准形式，这给出了一个完整判据：β_ĉ是双手性的当且仅当c是回文或具有奇周期，同时根据这些性质及非退化flype的存在性确定了精确的对称类型。
英文摘要
We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov--Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index~$3$. Every amphichiral knot of braid index~$3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word~$c$. Building on the Birman--Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on~$c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index~$3$, this yields a complete criterion: $\widehat{β_c}$ is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.
Comments4 figures, 27 pages. Comments welcome!
