  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.
