What is a virtual link?

Abstract

Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L ⊂ S ×I in a thickened, compact, oriented surface S such that the link complement (S × I) \ L has no essential vertical cylinder. AMS Classification 57M25; 57M27 57M15Keywords Virtual link, tetravalent graph, stable equivalence A virtual link L is an equivalence class of decorated, finite, tetravalent graphs Γ. The edges at each vertex must be cyclically ordered, and two opposite edges are labelled as an overcrossing, while the other two are labelled as an undercrossing. The equivalence relation is the one given by Reidemeister moves. The notion was proposed by Kauffman [5] in light of the fact that R-matrices and quandles, which are commonly used to make link invariants, also yield invariants of virtual links.We will borrow from the analysis of virtual links by Carter, Kamada, and Saito [1]. They show that virtual links are equivalent to stable equivalence classes of links projections onto compact, oriented surfaces. (Fenn, Rourke, and Sanderson have written a self-contained proof [2].) The surface S need not be connected, but we require that each component contains at least one component of the projection P . The projection P is considered up to Reidemeister moves, and the stabilization operation consists of adding a handle to S . Note that the feet of a stabilizing handle may lie in any two regions of the complement of P (Figure 1).As Figure 1 also shows, the reverse destabilization operation consists of cutting the surface S along a circle C which is disjoint from P , and capping the resulting boundary. If C separates the connected component of S of S containing it, then we require that both components of S \ C contain part of P ; otherwise destabilization would create a naked surface component.It is also well-known that a link drawn on a surface S , considered up to Reidemeister moves, is equivalent to a (tame) link L ⊂ S × I in the thickened surface

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“…A natural question about equivalence relations on classical links arises: the classical and the new one defined by means of generalized Reidemeister moves. As it turns out, these relations are the same; that is, virtual Reidemeister moves "do not spoil" the ordinary equivalence of knots (see [7,19] Let K 1 , K 2 be two non-intersecting diagrams of oriented virtual knots on the plane P with the property that some two-dimensional disc E intersects K 1 K 2 in two arcs AB ∈ K 1 and CD ∈ K 2 with opposite orientations; that is, the first arc is oriented from A to B, while the second is oriented from C to D. Here when one goes round the circle ∂E in a clockwise direction, the four points are encountered in the order A, B, C, D. We suppose further that there exists a line intersecting E and not intersecting the diagrams K 1 , K 2 , so that the diagrams K 1 and K 2 lie on different sides of . By the connected sum of the diagrams K 1 and K 2 (denoted by K 1 #K 2 ) we mean the diagram obtained from the diagram of the disjoint sum K 1 K 2 by removing the arcs AB and CD and adding the arcs DA and CB with the orientations of the diagrams K 1 and L 2 extended to the diagram K 1 #K 2 ; see Figure 6.…”

“…Virtual knots can also be interpreted topologically as knots in "thickened surfaces" S g ×I; the latter is the Cartesian product of a sphere with handles S g and the closed interval I, and the virtual knots are regarded to within isotopy and stabilization of these surfaces, that is, gluing and removal of new "thickened" handles which have empty intersection with the knot under consideration. In [19] Kuperberg showed that every virtual knot has a canonical minimal representation in S g × I with minimal g. 1 A corollary of Kuperberg's theorem is that the classical theory of knots can be embedded in the theory of virtual knots in the sense that two classical virtual knot diagrams are virtually equivalent if and only if they give isotopic knots in the usual sense. The first proof of this fact was given in [7].…”

Twisted link theory

“…When the surface being thickened is a sphere, the links are classical, however there are virtual links that are not classical links. Since by a result of Kuperberg [13], virtual links have unique irreducible representatives, virtual link theory is a proper extension of classical links.…”

“…By considering destabilization of the oriented thickening along annuli and Möbius bands in the complement of the link, we get an extension to links in oriented thickenings of a result of Greg Kuperberg [13] for virtual links.…”

Гомологии Хованова Виртуальных Узлов С Произвольными Коэффициентами

“…Важная теорема (см., например, [4]) состоит в том, что виртуальные узлы -это как раз узлы в утолщенных поверх-ностях, рассматриваемые с точностью до изотопий, а также гомеоморфизмов утолщенных поверхностей S g × R 1 , сохраняющих координату вдоль R 1 , и ста-билизаций/дестабилизаций поверхностей. Ключевой является теорема Г. Ку-перберга [5] о том, что минимальный (по количеству ручек поверхности) пред-ставитель заданного класса виртуального узла единствен. Из этого следует, что классическая теория узлов является частью виртуальной теории узлов, т.е.…”