Computation of the Kauffman Bracket Skeleton

Abstract

In this paper, we present an invariant called the Kauffman bracket skeleton, which is a simplification of the generalized Kauffman bracket polynomial in two variables. The idea is to take into account only the order and values of coefficients and disregard the degrees of one of the variables. However, the proposed simplification is more compact, and at the same time is not weaker than the original generalized Kauffman bracket polynomial in the sense of, for example, tabulation of prime knots and links in the thickened torus up to complexity $4$ inclusively. In order to confirm this fact, we present tables of the proposed invariant for the tabulated prime knots and links in the thickened torus. Also, we construct an algorithm to compute the proposed simplification. The algorithm does not require to work with degrees of the disregarded variable and uses a symmetry of the Kauffman bracket presented by rows of the alternating Pascal's triangle. Finally, we give a remark on an interpretation of the proposed invariant and algorithm in the case of classical knots and links.