Annular and pants thrackles

A thrackle is a drawing of a graph in which each pair of edges meets precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of a graph on the plane cannot have more edges than vertices. We prove the Conjecture for thrackle drawings all of whose vertices lie on the boundaries of $d \le 3$ connected domains in the complement of the drawing. We also give a detailed description of thrackle drawings corresponding to the cases when $d=2$ (annular thrackles) and $d=3$ (pants thrackles).Comment: 17 page

All odd musquashes are standard

AbstractWe prove that there are no n-agonal musquashes for n odd, apart from the standard ones. This completes the classification of musquashes

ALL ODD MUSQUASHES ARE STANDARD

Abstract. We prove that there are no n-agonal musquashes for n odd, apart from the standard ones. This completes the classification of musquashes. An n-agonal musquash is a planar drawing of the n-gon, with consecutive edges e1,...,en, such that each pair of edges meets precisely once, either at a vertex or at a transverse crossing, and if edge e1 intersects edges in the following order: ek1,...,ekn−3, then for all i =2,...,n, edge ei intersects edges in the following order: ek1+i−1,...,ekn−3+i−1, where the edge subscripts are computed modulo n. For n odd, the standard n-agonal musquash has the n-th roots of unity as its vertices and for each k, edge ek goes from e (k−1)(1−n)πi/n to e k(1−n)πi/n. Figure 1. The standard pentagonal musquash This paper is the sequel to [2], in which we showed that there are no n-agonal musquashes with n even and n>6. In this present paper we exploit the same ideas in order to complete the classification of musquashes. We maintain the notation and terminology of [2]. By stereographic projection, we regard musquashes as being drawn on the 2-sphere S 2. We regard two musquashes as being equivalent if one can be obtained from the other by a homeomorphism of S 2; this equivalence is also called geotopy [3]. Theorem. For n odd, every n-agonal musquash is equivalent to the standard one. Before proving this theorem, let us make some remarks concerning the notion of equivalence. First, recall that given a musquash M, and an orientation of M