Morwen Thistlethwaite

Thistlethwaite unknot

Morwen B. Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory.

Biography

Morwen Thistlethwaite received his BA from the University of Cambridge in 1967, his MSc from the University of London in 1968, and his PhD from the University of Manchester in 1972 where his advisor was Michael Barratt. He studied piano with Tanya Polunin, James Gibb and Balint Vazsonyi, giving concerts in London before deciding to pursue a career in mathematics in 1975. He taught at the North London Polytechnic from 1975 to 1978 and the Polytechnic of the South Bank, London from 1978 to 1987. He served as a visiting professor at the University of California, Santa Barbara for a year before going to the University of Tennessee, where he currently is a professor. Thistlethwaite's son Oliver is also a mathematician.^[1]

Work

Tait conjectures

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)^[2]

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

Thistlethwaite's algorithm

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 groups of cube positions that can be solved using a certain set of moves. The groups are:

• G₀ = <L,R,F,B,U,D>

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

• G₁ = <L,R,F,B,U2,D2>

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.

• G₂ = <L,R,F2,B2,U2,D2>

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.

• G₃ = <L2,R2,F2,B2,U2,D2>

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

• G₄ = {I}

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 likely 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.^[3]

Dowker notation

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

See also

• Optimal solutions for Rubik's Cube

References

External links

• http://www.math.utk.edu/~morwen/ - Morwen Thistlethwaite's home page.
• Morwen Thistlethwaite at the Mathematics Genealogy Project