Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case

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: β_ĉ 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.

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BibTeX RIS

BibTeX

@misc{jung2026amphichiral,
  title = {{Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case}},
  author = {Hyungseok Jung},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12295},
  eprint = {2609.12295},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12295},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case
AU  - Hyungseok Jung
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.12295
UR  - https://arxiv.org/abs/2609.12295
AB  - 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: β_ĉ 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.
ER  -

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