[2206.11379] Number of sticks in rail arc

Download a PDF of the paper entitled The stick number of rail arcs by Nicholas Cazet

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overview: Consider two parallel lines $\ell_1$ and $\ell_2$ in $\mathbb{R}^3$. A rail arc is a $\mathbb{R}^ Embedding of arcs in 3$. \ell_2$. Rail arcs are considered rail isotopes, up to surrounding isotopes of $\mathbb{R}^3$, and each self-homologous isotope maps $\ell_1$ and $\ell_2$ to itself. When manifolds and maps are in the piecewise linear category, these rail arcs are called stick rail arcs.

The number of sticks for a rail arc class is the minimum number of sticks, which are line segments within the PL arc, required to create a representative. In this paper, we compute stick numbers for rail arc classes with a maximum number of crossings of 2, and use the winding number invariant to compute stick numbers for infinitely many rail arc classes.

Each rail arc class has two canonically associated knot classes (under companion and over companion). This paper also introduces knot class rail stick numbers. This is the minimum number of sticks required to create a rail arc, the companion below or above which is the knot class. Rail stick counts are calculated for all knot classes with a maximum of 9 crossings. The number of sticks for multi-component rail arc classes and the number of lattice sticks for rail arcs are taken into account.