Bisected vertex leveling of plane graphs: braid index, arc index and delta diagrams

TL;DR

This paper introduces a new planar embedding technique called bisected vertex leveling for plane graphs, providing elementary proofs for bounds on braid and arc indices of knots and links, and establishing a quadratic bound for delta diagrams.

Contribution

It presents a novel bisected vertex leveling method and uses it to give simplified proofs of known bounds and new bounds on delta diagrams for knots and links.

Findings

Elementary proofs of bounds on braid and arc indices.

Quadratic upper bound for crossing number of delta diagrams.

Introduction of bisected vertex leveling technique.

Abstract

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $\alpha(L)$ for any knot or non-split link $L$, which are $b(L) \leq \frac{1}{2} c(L) + 1$ and $\alpha(L) \leq c(L) + 2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.