## A Much Faster Way to check a Notation's "Drawability"

To summarise what was proved in the previous page: a notation is drawable if and only if the shadow it would denote does not violate the Jordan Curve Theorem. This requires checking up to 3^{2N}/2 pairs of loops.

There is a much faster way however to check a notation's drawability. Instead of considering each possible notation and doing the full calculations on all pairs of loops, one performs Equal Drawability moves on the notation. In other words, one replaces the notation under consideration by another notation such that either both are drawable or none of them. It is possible (in fact almost certain) that by performing such moves one eventually arrives to a notation that has already been checked. By applying such a procedure the whole program was accelerated by a factor of a hundred.

The equal drawability moves should not be confused with the reidemeister moves; while the latter are performed on the full projection, the former are performed on the shadow. And while the latter replace a knot with a topologically equivalent one, the former replace a shadow not by an equivalent one, but merely by one that is as drawable as the previous one. There are three kinds of "equal drawability" moves:

• Remove a pair {i,i+1} and decrease all numbers higher than i+1 by 2.
• If the notation consists of pairs {i,j}, {i+1,j+1}, {i+2,j+2}, where i < j , remove the latter two pairs; all numbers between i+2 and j decrease by 2, all numbers higher than j+2 decrease by 4. Similarly if the notation consists of {i,j+2}, {i+1,j+1}, {i+2,j}.
• If the notation consists of pairs {i,j}, {i',k}, {j',k'}, where |i'-i|=|j'-j|=|k'-k|=1, replace them with {i,k'}, {i',j'}, {j,k}.
• The first two moves always lead to a notation with a smaller crossing number, and thus if a notation allows for such moves, one has definitely avoided the time consuming full calculation. The latter may or may not be halpful; when the crossing number however becomes large, the third move possibilities increase, and thus it is very likely that one may arrive to a notation whose drawability has already been found.

To see how a shadow is transformed under these moves, click here. From the figures one should easily understand why these moves lead to equally drawable notations.