9401139 Thistlethwaite Using the vastly improved computer power now available on workstations, the investigator has recently classified knots up to 14 crossings. He will continue the compilation of knots through 15 or 16 crossings. This computational work is being done jointly with J. Hoste (Pitzer College). The investigator will endeavor to answer various theoretical problems which arise naturally from the classification process. In particular, he will seek to complete the computation of the mapping class group of an alternating link pair, and he will look into questions concerning primality of knots and questions concerning homomorphic images of knot groups or subgroups of knot groups. He will search for a second proof of the Tait Conjecture and for possible extensions of this conjecture. These extensions could be used to answer conjectures concerning the asymptotic density of alternating knots. Thistlethwaite's research treats the topology of knots and links. A knot is a simple closed curve, which one may think of as a closed loop of string, situated somehow in space; a link is a more general kind of object, consisting of one or more separate loops. The mathematical interest lies not so much in the intrinsic loop, but rather in the way in which it is situated in 3-dimensional space, i.e. in the way in which it might be "knotted." Topologists consider that two knots (or links) are equivalent if it is possible to maneuver one to the other by bending or stretching; no cutting is allowed. One of the fascinating aspects of knot theory is that it contains many problems which are easy to state and easy for the layman to understand, yet which persistently resist all attempts at solution. One such problem, which William Menasco (SUNY Buffalo) and Thistlethwaite solved last year, was the hundred-year old Tait conjecture on alternating links. The comprehensive table of knots up to 16 crossings which Thistlethwaite and Hoste are currently producin g will be used to test and formulate other conjectures. In addition, Thistlethwaite will investigate the unknotting number of an alternating knot, this being the minimal number of times it is necessary to pass a knot through itself to render it trivial. His search for an alternative proof of the Tait conjecture will involve "non-Euclidean" geometry. ***
