Meander diagrams of knots and spatial graphs: Proofs of generalized Jablan–Radović conjectures

Abstract

We study decomposition into simple arcs (i.e., arcs without self-intersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radović stating that each knot has a meander diagram, i.e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radović stating that each 2-bridge knot has a semi-meander minimal diagram, i.e., a minimal diagram composed of two simple arcs.