论文标题
铁轨弧的棒数
The stick number of rail arcs
论文作者
论文摘要
考虑两个并行行$ \ ell_1 $和$ \ ell_2 $ in $ \ mathbb {r}^3 $。铁路弧是$ \ mathbb {r}^3 $中的弧的嵌入，使一个端点在$ \ ell_1 $上，另一个端点在$ \ ell_2 $上，其内部与$ \ ell_1 \ cup \ cup \ ell_2 $脱节。铁路弧被视为铁路同位素，每个自我塑料映射$ \ ell_1 $和$ \ ell_2 $的$ \ mathbb {r}^3 $的环境同位素。当在分段线性类别中拍摄歧管和地图时，这些导轨弧被称为棒轨弧。 轨道弧类的棍子数量是木棍的最小数量，P.L中的线段。弧，需要创建代表。本文将计算出最多有交叉数的铁路ARCS类的棒数，并使用绕组数量不变来计算无限多个铁路弧类的棍子数量。 每个铁轨弧班都有两个规范相关的结班，包括同伴的下和伴侣。本文还引入了轨道棒的结构，这是创建轨道弧线所需的最小棍子数量，该弧线下方或伴随的是结式类别。轨道棒号码是针对所有打结类计算的，最多最多为9个。多组件铁路弧类的棒数被认为是轨道杆的晶格杆数。
Consider two parallel lines $\ell_1$ and $\ell_2$ in $\mathbb{R}^3$. A rail arc is an embedding of an arc in $\mathbb{R}^3$ such that one endpoint is on $\ell_1$, the other is on $\ell_2$, and its interior is disjoint from $\ell_1\cup\ell_2$. Rail arcs are considered up to rail isotopies, ambient isotopies of $\mathbb{R}^3$ with each self-homeomorphism mapping $\ell_1$ and $\ell_2$ onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs are called stick rail arcs. The stick number of a rail arc class is the minimum number of sticks, line segments in a p.l. arc, needed to create a representative. This paper will calculate the stick numbers of rail arcs classes with a crossing number at most 2 and use a winding number invariant to calculate the stick numbers of infinitely many rail arc classes. Each rail arc class has two canonically associated knot classes, its under and over companions. This paper also introduces the rail stick number of knot classes, the minimum number of sticks needed to create a rail arcs whose under or over companion is the knot class. The rail stick number is calculated for all knot classes with crossing number at most 9. The stick number of multi-component rail arcs classes is considered as well as the lattice stick number of rail arcs.
