The stick number of rail arcs

TL;DR

This paper calculates the minimum number of line segments needed to construct certain rail arcs and knots, introducing new invariants and extending results to classes with higher crossing numbers.

Contribution

It introduces the rail stick number for knot classes and computes it for all knots with crossing number up to 9, extending previous work on rail arcs.

Findings

Calculated stick numbers for rail arcs with crossing number ≤ 2

Introduced rail stick number for knot classes and computed for knots up to 9 crossings

Extended analysis to multi-component and lattice rail arcs

Abstract

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.