Spindle-configurations of skew lines

Abstract

We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindlestructure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations.We also slightly simplify the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration.Finally, we describe an algorithm which constructs a spindle in a given switching class, or proves non-existence of such a spindle.Corresponding author.

Eulerian partitions for configurations of skew lines

“…Isotopy classes of spindle-configurations are well understood by the following combinatorial description, given in [1]: Theorem 1.1. Two spindle-permutations σ, σ ′ give rise to isotopic spindle-configurations if and only if σ and σ ′ are spindle-equivalent.…”

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In this paper, which is a complement of [1], we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration.We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindlepermutation.A spindle (or isotopy join or horizontal configuration) is a particularly nice configuration of skew lines in which all lines intersect an oriented additional line A, called the axis of the spindle.

…”

“…In this paper, which is a complement of [1], we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration.…”