Generalizations of the Kauffman polynomial for knots in the thickened surface of genus 2

In this paper, we discuss two invariants, which are results of different approaches to the generalization of the Kauffman polynomial for the case of knots in the thickened surface of genus 2. The first generalization proposes to distinguish only two types of curves (cut and noncut) and seems to be rather strong to prove non equivalence of a big number of knots in the thickened surface of genus 2. However, the first generalization provides no tools to determine if a knot can be realised in the thickened surface having a smaller genus. In its turn, the second generalization proposes in addition to distinguish isotopy types of noncut curves. As a result, such an invariant is enough to perform the destabilisation, if it is possible, by the following steps. First, determine a destabilisation curve or prove that such a curve does not exist. Second, if a destabilisation curve exists, use a sequence of elementary transformations to reduce the found destabilisation curve to the standard form. Finally, reduce isotopy types of noncut curves involved in the terms of the generalized Kauffman polynomial to obtain the generalized Kauffman polynomial of the same knot realised in the thickened surface having a smaller genus. Computational examples show the effectiveness of the second generalization and the weakness of the first generalization in the destabilisation.