2014/06/09 by Slavik Jablan, Louis Kauffman, Jablan, Slavik +4
Computer Science · Mathematics · #57M27 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:57M27 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1406.2378
21 pages, 15 figures
arxiv created 2014/06/09 · openalex publication_date 2014/06/09 · arxiv updated 2014/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.