论文标题
超级桥指数的花瓣数的花瓣数
Petal number of torus knots using superbridge indices
论文作者
论文摘要
打结$ k $的花瓣投影是一个结的投影，它由单个多跨和非巢环组成。由于花瓣投影给定结的自然数序列，因此花瓣投影是研究结理论的有用模型。众所周知，每个结都有花瓣投影。花瓣数$ p（k）$是表示结$ k $作为花瓣投影所需的最小循环数。在本文中，我们发现了超级桥指数与花瓣数的任意结之间的关系。通过使用此关系，我们找到了$ t_ {r，s} $的花瓣编号，如下所示； $$ p（t_ {r，s}）= 2S-1 $$当$ 1 <r <s $和$ r \ equiv 1 \ mod s-r $。此外，我们还找到了$ t_ {r，s} $的花瓣编号的上限，如下所示； $ p（t_ {r，s}）\ leq2s- 2 \ big \ lfloor \ frac {s} {r} {r} \ big big \ rfloor +1 $ 1 $ s \ equiv \ equiv \ pm 1 \ mod r $。
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 $T_{r,s}$ as follows; $$p(T_{r,s})=2s-1$$ when $1 < r < s$ and $r \equiv 1 \mod s-r$. Furthermore, we also find the upper bound of the petal number of $T_{r,s}$ as follows; $$p(T_{r,s})\leq2s- 2\Big\lfloor \frac{s}{r} \Big\rfloor +1$$ when $s \equiv \pm 1 \mod r$.
