I am trying to write a computer program to solve Rubik's cubes using a version of Thistlethwaite's algorithm. Instead of using lookup tables to perform each phase of the algorithm, I'm using a breadth-first search. However, this requires me to recognize when I have reached the goal for each phase and can successfully transition into the next phase.
I'm having trouble identifying what marks the end of phase 3. This website (http://www.jaapsch.net/puzzles/thistle.htm) claims that phase 3 is over when "edges in the L and R faces are placed in their correct slices, the corners are put into their correct tetrads, the parity of the edge permutation (and hence the corners too) is made even, and the total twist of each tetrad is fixed". I understand edge slices, tetrads, and parity; but I couldn't find any references explaining what the "twist of a tetrad" is.
Can anyone shed some light on this term, or suggest some other way to tell when phase 3 of the algorithm is over?