Narrow Results

We introduce the virtual magnetic Kauffman bracket skein module, and show how to find a skein relation for the f-polynomial through the module. Furthermore, we give a unified approach to constructions of skein relations which hold under certain conditions.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³. In this paper, we define the Gordian complex of virtual knots which is a simplicial complex whose vertices consist of all virtual knots by using the local move which makes a real crossing a virtual crossing. We show that for any virtual knot K₀ and for any given natural number n, there exists a family of virtual knots {K₀, K₁,…,K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the Gordian distance of virtual knots d_v(K_i, K_j) = 1. And we also give a formula of the f-polynomial for the sum of tangles of virtual knots.

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial. For any pair of virtual knots $K$ and $M$ with $n$-dwrithe $\nabla J_n(K)\ne 0$ we construct an infinite family of different connected sums of $K$ and $M$ which can be distinguished by $F$-polynomials.

As a generalization of the classical knots, knotoids deal with the open ended knot diagram in a surface. In recent years, many polynomial invariants for knotoids appeared, such as the bracket polynomial, the index polynomial and the $n$th polynomial. In this paper, we introduce a new polynomial invariant $F$-polynomial for knotoids and discuss some properties of the $F$-polynomial. Finally, we construct a family of knotoid diagrams which can be distinguished from each other by the $F$-polynomial but cannot be distinguished by the index polynomial and the $n$th polynomial.