Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Jung, Hyungseok
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212592710