English

Petal diagram from simple braids

Geometric Topology 2023-10-17 v1

Abstract

We construct petal diagrams from simple braids. This approach allows us to confirm a conjecture proposed by Kim, No and Yoo, which states that the petal number of the nontrivial torus knot Tr,sT_{r,s} (r<sr<s) is at most 2s2sr+12s-2\lfloor\frac{s}{r}\rfloor+1. As a consequence, we deduce that the petal number of a nontrivial torus knot Tr,sT_{r,s} is equal to 2s12s-1 if and only if r<s<2rr<s<2r.

Keywords

Cite

@article{arxiv.2310.09569,
  title  = {Petal diagram from simple braids},
  author = {Zipei Nie},
  journal= {arXiv preprint arXiv:2310.09569},
  year   = {2023}
}

Comments

8 pages, 4 figures

R2 v1 2026-06-28T12:50:38.379Z