Meander diagrams of knots and spatial graphs: proofs of generalized Jablan--Radovi\'{c} conjectures

TL;DR

This paper proves that certain spatial graphs and knots can be represented with diagrams composed of simple, non-self-intersecting arcs, confirming conjectures about meander diagrams and their minimal forms.

Contribution

It generalizes the conjecture that all knots have meander diagrams and proves that 2-bridge knots have minimal semi-meander diagrams, advancing understanding of knot diagram representations.

Findings

Existence of simple arc decompositions for spatial graphs without knotted loops.

Confirmation that all knots admit meander diagrams.

Proof that 2-bridge knots have minimal semi-meander diagrams.

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. Radovi\'{c} 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 Radovi\'{c} stating that each 2-bridge knot has a semi-meander minimal diagram, i. e., a minimal diagram composed of two simple arcs.