A NOTE ON THE UTILITY OF KAUFFMAN'S STATE SUMMATION FORMULA FOR THE BRACKET POLYNOMIAL

Abstract

Kauffman's state summation formula is especially useful for computing the bracket polynomials of projection diagrams which are related by smoothings or crossing changes. This facilitates the writing of a symbolic algebra program which computes the normalized bracket polynomials and frequencies of knots and links whose projection diagrams result from a given knot's oriented projection diagram either by crossing changes or by orientation preserving smoothings called natural smoothings. These frequencies provide insight into the unknotting game (and similar resultant games) whose object is to specify crossing changes or natural smoothings that will transform a given projection diagram of a knot into a projection diagram representing an unknot (or some other specified knot or link). The practical utility of the state summation formula is greatly enhanced by means of diagrams for closed tangle sums. These diagrams offer a special cost-reducing method to obtain crucial information needed to compute the state summation formula. This special method also gives insight into why the bracket is unchanged by mutation and contributes a strategy to the enigmatic search for a non-trivial knot with Jones polynomial equal to one.