The Delunification Process and Minimal Diagrams

TL;DR

This paper proves that any colored link diagram can be transformed into a lune-free diagram with the same number of colors, introduces grey sets for better bounds on minimal colors, and computes related invariants for small links.

Contribution

It establishes the delunification process, introduces grey sets for lower bounds, and computes minimal color and crossing numbers for prime links up to 16 crossings.

Findings

Any colored link diagram is equivalent to a lune-free diagram with the same number of colors.

Higher lower bounds for minimal colors are obtained using grey sets and computational methods.

Lune-free crossing numbers are calculated for links with up to 16 crossings.

Abstract

A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same number of colors. Thus any colored link diagram with a minimum number of colors (known as a minimal diagram) is equivalent to a colored lune-free diagram with that same number of colors. We call the passage from a link diagram to an equivalent lune-free diagram its delunification process. We then introduce a notion of grey sets in order to obtain higher lower bounds for minimum number of colors. We calculate these higher lower bounds for a number of prime moduli with the help of computer programs. For each number of crossings through 16, we list the lune-free diagrams and we color them. If the number of colors equals the corresponding higher lower bound we know we have a minimum number of colors. We also introduce and list the lune-free crossing number of a link i.e., the minimum number of crossings needed for a lune-free diagram of this link, and other related link invariants.