2022/09/29 by Hyoung-Jun Kim, Kim, Hyoungjun, Sungjong No +3
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2209.14702
openalex publication_date 2022/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A petal projection of a knot K is a projection of a knot which consists of a single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number p(K) is the minimum number of loops required to represent the knot K as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of Tr,s as follows; p(Tr,s)=2s-1 when 1 < r < s and r ≡ 1 \mod s-r. Furthermore, we also find the upper bound of the petal number of Tr,s as follows; p(Tr,s)≤2s- 2\lfloor (s)/(r) \rfloor +1 when s ≡ ± 1 \mod r.