TABULATION OF PRIME LINKS IN THE THICKENED SURFACE OF GENUS 2 HAVING DIAGRAMS WITH AT MOST 4 CROSSINGS

Abstract

The aim of this paper is to tabulate all prime not oriented links in the thickened surface of genus 2 having diagrams with no more than 4 crossings. A preliminary set of diagrams is constructed based on the table of prime link projections in the  surface of genus 2. In order to remove duplicates and prove that all the rest links  are not equivalent, as well as to prove that all tabulated links admit no destabilisations, we use an invariant called the Kauffman bracket frame, which is  a simplification of the generalized Kauffman bracket polynomial. The idea of the invariant is to consider only the values  and order of coefficients and do not take into account the powers of one of the variables. Finally, we prove that each tabulated link can not be given by a connected sum under the hypothesis that the sum of complexities of the terms that form the connected sum is not more than  the complexity of the connected sum.