Mathematical Sciences: Connections Between Geometry and LinkPolynomials

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Thistlewaite plans to examine several interconnected problems in classical knot theory and the theory of `trace' polynomial invariants. The principal task is the search for intrinsic geometrical definitions of the new polynomial invariants of links. Also, Thistlewaite will seek improved combinatorial definitions of the two 2-variable generalizations of the Jones polynomial, from which it is hoped that rapid proofs of the existence of these polynomials will materialize. On a purely geometrical front, he will pursue the question as to whether crossing-number of links is additive with respect to connected sum.