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

The Delunification Process and Minimal Diagrams