The stick number of rail arcs

Abstract

Consider two parallel lines [Formula: see text] and [Formula: see text] in [Formula: see text]. A rail arc is an embedding of an arc in [Formula: see text] such that one endpoint is on [Formula: see text], the other is on [Formula: see text], and its interior is disjoint from [Formula: see text]. Rail arcs are considered up to rail isotopies, ambient isotopies of [Formula: see text] with each self-homeomorphism mapping [Formula: see text] and [Formula: see text] 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 calculates the stick number of rail arcs classes with a crossing number at most 2 and uses 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 29 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.