Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Abstract

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [Formula: see text]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [Formula: see text]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

“…For q 10, polynomials F n VK q (t, ) were computed in [21]. It turns out that each of the polynomials F 2 (t, ), F 4 (t, ) and F 6 (t, ) is able to distinguish virtual knots VK 1 , VK 2 and VK 3 but not able to distinguish virtual knots VK 3 , .…”

“…, q, are repeating blocks separated by dashed lines. These virtual knots were constructed in [21] to provide examples of q-simplexes in the Gordian complex of virtual knots corresponding to the arc shift move defined in [25]. It was shown in [25] that the arc shift move together with generalized Reidemeister moves is an unknotting operation for virtual knots.…”

“…It was shown in [25] that the arc shift move together with generalized Reidemeister moves is an unknotting operation for virtual knots. By using the Kauffman F-polynomial, it was shown in [21] that virtual knots VK q are distinct for distinct q.…”

“…Two types of classical crossings are presented in Figure 1. F-polynomials were calculated for tabulated virtual knots in [19] and [20], and successfully used to distinguish some oriented virtual knots in [21]. Another approach to construct invariants of flat virtual knots can be based on the representation of flat virtual braids by automorphisms of free groups; see, for example, ref.…”

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

“…Two types of classical crossings are presented in Figure 1. F-polynomials were calculated for tabulated virtual knots in [8] and [19], and successfully used to distinguish some oriented virtual knots in [6]. Another approach to construct invariants of flat virtual knots can be based on representation of flat virtual braids by automorphisms of free groups, see, for example, [1].…”