Weak rectangular diagrams, multi-crossing number, and arc index

TL;DR

This paper establishes a new inequality relating the arc index and multi-crossing numbers of links, introduces weak rectangular diagrams, and shows how to convert loose diagrams to rectangular ones without changing the arc index.

Contribution

It generalizes known inequalities by relating arc index and multi-crossing numbers and introduces the concept of weak rectangular diagrams for link representations.

Findings

Proves the inequality L - 2   c_2(D) + \u2212   abla_{n>2} (2n-4) c_n(D) for links.

Introduces weak rectangular diagrams and shows conversion to rectangular diagrams preserves arc index.

Generalizes and subsumes many known inequalities related to multi-crossing numbers.

Abstract

For a non-split multi-crossing diagram $D$ of a link $L$ we show that $\alpha(L)-2 \leq c_2(D) + \sum_{n> 2}(2n-4)c_n(D)$ holds. Here $\alpha(L)$ is the arc index and $c_n(D)$ is the number of $n$-crossings of $D$. This generalizes and subsumes many known inequalities related to multi-crossing numbers. In the course of proof, we introduce a notion of weak rectangular diagram and show that a loose rectangular diagram can be converted to usual rectangular diagram preserving its arc index.