CLASSIFICATION OF PRIME KNOTS IN THE THICKENED SURFACE OF GENUS 2 HAVING DIAGRAMS WITH AT MOST 4 CROSSINGS

Abstract

The goal of this paper is to tabulate all prime knots in the thickened surface of genus 2 having diagrams with at most 4 crossings. First, we introduce definition of prime  knot in the thickened  surface of genus 2. Second, we construct a table of prime knots. To this end, we use the table of prime knot projections in the  surface of genus 2  to construct a preliminary set of diagrams. In order to remove duplicates and prove that all the rest knots  are inequivalent, as well as to prove that all tabulated knots admit no destabilisations, we propose an invariant called the Kauffman bracket frame, which is  a simplification of the generalized Kauffman bracket polynomial. 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 up to complexity  4 inclusively. Finally, we prove that each tabulated knot can not be represented as a connected sum under the hypothesis that the complexity of a connected sum is not less than the sum of complexities of the terms that form the  sum.