Morwen Thistlethwaite

Morwen Thistlethwaite helped prove the Tait conjectures, which are:

1. Reduced alternating diagrams have minimal link crossing number.
2. Any two reduced alternating diagrams of a given knot have equal writhe.
3. Given any two reduced alternating diagrams D₁,D₂ of an oriented, prime alternating link, D₁ may be transformed to D₂ by means of a sequence of certain simple moves called flypes. Also known as the Tait flyping conjecture.
   (adapted from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/TaitsKnotConjectures.html)[3]

Morwen Thistlethwaite, along with Louis Kauffman and Kunio Murasugi proved the first two Tait conjectures in 1987 and Thistlethwaite and William Menasco proved the Tait flyping conjecture in 1991.

Thistlethwaite also came up with a famous solution to the Rubik's Cube. The way the algorithm works is by restricting the positions of the cubes into a subgroup series of cube positions that can be solved using a certain set of moves. The groups are:

• ${\displaystyle G_{0}=\langle L,R,F,B,U,D\rangle }$

This group contains all possible positions of the Rubik's Cube.

• ${\displaystyle G_{1}=\langle L,R,F,B,U^{2},D^{2}\rangle }$

This group contains all positions that can be reached (from the solved state) with quarter turns of the left, right, front and back sides of the Rubik's Cube, but only double turns of the up and down sides.

• ${\displaystyle G_{2}=\langle L,R,F^{2},B^{2},U^{2},D^{2}\rangle }$

In this group, the positions are restricted to ones that can be reached with only double turns of the front, back, up and down faces and quarter turns of the left and right faces.

• ${\displaystyle G_{3}=\langle L^{2},R^{2},F^{2},B^{2},U^{2},D^{2}\rangle }$

Positions in this group can be solved using only double turns on all sides.

• ${\displaystyle G_{4}=\{1\}}$

The final group contains only one position, the solved state of the cube.

The cube is solved by moving from group to group, using only moves in the current group, for example, a scrambled cube always lies in group G₀. A look up table of possible permutations is used that uses quarter turns of all faces to get the cube into group G₁. Once in group G₁, quarter turns of the up and down faces are disallowed in the sequences of the look-up tables, and the tables are used to get to group G₂, and so on, until the cube is solved.[4]

Thistlethwaite, along with Clifford Hugh Dowker, developed Dowker–Thistlethwaite notation, a knot notation suitable for computer use and derived from notations of Peter Guthrie Tait and Carl Friedrich Gauss.