Optimal Void Cube Up Face Odd Parity Maneuvers

When solving the void cube one encounters odd position parity positions as the cube approaches being solved. The last step for me is to solve the Up face corners and this is where I encounter odd permutations. Corners first solvers may encounter an odd permutation of the Up face edges. The question is what is the shortest maneuver which will convert an odd permutation of the Up face corners into an even permutation of the Up face corners leaving everything else unchanged. This is regardless of the effect on the corner twist. Likewise, what is the shortest maneuver which will convert an odd parity up face edge permutation to an even parity permutation regardless of the edge flip?

There are (4! x 3³) / 2 = 324 odd parity Up face corner position and (4! x 2³) / 2 = 96 odd parity Up face edge positions. These reduce by C4v symmetry to 53 symmetry unique corner positions and 21 symmetry unique edge positions. I have found optimal solutions for these positions and determined that odd parity edge positions may be corrected in a minimum of 12 q-turns or 8 f-turns. Odd parity corner permutations may be corrected in a minimum of 15 q-turns or 13 f- turns.

Two turn sequences are given for each maneuver above. The first in the standard < R U F L D B > turn set and the second in the equivalent < R U F TR TU TF > turn set. In the second, tier turns are substituted for the L D B turns. Rather than turning say the L face one turns the MR slice and the R face in the opposite direction. As discussed elsewhere, in the second system turn sequences behave as elements of a mathematical group. This is highlighted by inverting the turn sequence. Inverting the void cube sequence has the expected result—the inverse permutation is applied to the Up face. Inverting the standard sequence has unexpected results. The standard sequences of necessity give the void cube rotated by an odd number of 90° whole cube rotations. In order for the inverse standard turn sequence to give the expected result one must rotate the void cube by this rotation prior to applying the inverse turn sequence.

The optimal edge maneuvers are all figure eight four cycle swaps with different edge flip. The single optimal f-turn maneuver gives the same permutation as the second q-turn maneuver. The corner maneuvers include both single swaps and four cycle swaps. The second q-turn maneuver, equivalent to the first f-turn maneuver, is of particular interest. This is a clockwise four cycle of the Up face corners with no concomitant twist.