target="_blank" rel="nofollow" href="#fb3_img_img_4e07e263-2c58-52a9-bc6c-87497a69abf7.png"/> into Σ × [0, 1] considered up to isotopies.
As in the classical case, links in the thickening of the surface Σ can be presented by their diagrams — 4-valent graphs embedded into Σ whose vertices have undercrossing-overcrossing structure. The equivalence of the diagrams is generated by diagram isotopies and Reidemeister moves (Fig. 3.2).
Given a link L in Σ × [0, 1], a stabilisation operation is defined by the attaching a thickened handle along a pair of annuli C1 × [0, 1] and C2 × [0, 1] which do not intersect the link, see Fig. 3.15. The result is a link in the thickening of a surface of higher genus. The inverse operation is called destabilisation.
Fig. 3.15Stabilisation
N. Kamada and S. Kamada [Kamada and Kamada, 2000] showed that virtual links can be defined as equivalence classes of pairs (Σ, L), where Σ is an oriented closed surface and L is a link in the thickening of Σ, modulo isotopies of L, natural isomorphisms of these pairs and stabilisations/destabilisation.
Classical knots and links can be considered as links in the thickening of the sphere S2 .
Many invariants of classical knots can be straightforwardly extended to virtual knots. For example, the Kaufman bracket (3.1) is invariant under virtual moves on virtual diagram, so Jones polynomial is an invariant of virtual knots. In some cases the Jones polynomial shows that a virtual diagram defines a non classical link.
Remark 3.1. Historically, virtual knots and links were defined first by L. H. Kauffman in [Kauffman, 1997] as equivalence classes of plane diagrams with virtual crossings. Later M.N. Goussarov, M.B. Polyak and O. Ya. Viro in [Goussarov, Polyak and Viro, 2000] introduced moves on Gauß diagrams and showed their theory was equivalent to Kauffman’s virtual knots. And finally, N. Kamada and S. Kamada [Kamada and Kamada, 2000] proposed an approach to virtual knots that uses thickenings of surfaces and the stabilisation.
The fact that virtual knots extend classical knots (more precisely, that the natural map from classical knots to virtual ones is injective) was established by G. Kuperberg [Kuperberg, 2002]. Further the fact was reproved many times. A proof based on the parity theory is given in Section 5.6.1.
Given a virtual link diagram, forgetting undercrossing-overcrossing structure at the vertices of the diagram yields a flat link diagram. In other words, flat diagram is an equivalence class of link diagrams modulo crossing switches, see Fig. 3.16.
Fig. 3.16Crossing switch
Equivalence classes of flat link diagrams modulo classical and virtual Reidemeister moves are called flat links.
Given a closed surface Σ, the flat knots on it can be identified with the homotopy classes of free loops in the surface Σ. This identification implies all flat classical knots (i.e. free loops in the sphere) are trivial.
Further simplification of knot structure leads to the notion of free knots and links. A free link is an equivalence class of a virtual link diagram modulo Reidemeister moves, crossing switchs and the virtualisation, see Fig. 3.17.
Fig. 3.17Virtualization
On the other hand, free knots can be defined as equivalence classes of chord diagrams modulo Reidemeister moves given in Fig. 3.18. Comparing with Gauß diagrams, chords here do not have orientation nor marks.
Fig. 3.18Reidemeister moves of chord diagrams
3.2Curve reduction on surfaces
Let us fix a two-dimensional manifold Σ. For simplicity we assume Σ to be oriented and closed, i.e. to be the sphere with g handles, g ≥ 0.
We are interested in homotopy classes of curves in Σ, that are also called free loops.
From the topological point of view these curves correspond to conjugacy classes of the group π1 (Sg). We denote the set of homotopy classes of curves in Σ as
We shall consider immersions S1 → Σ in general position, i.e. immersions such that all the points having more than one preimage are transverse double points, i.e. points of transverse intersections of the images of two arcs in S1.
Note that any curve can be approximated by a curve in general position. Since curves, which are close to each other, represent the same homotopy class, we can consider immersions in general positions as representatives of the free loops.
A diagram D in a closed surface Σ is a 4-graph Γ embedded in Σ, so that the complement Σ\Γ is a union of two dimensional cells. The graph Γ is called the diagram graph of D.
Analogously, one can define a diagram in a non compact surface or surface with boundary.
We shall not distinguish immersions which are combinatorially equivalent. We shall call such immersions diagrams.
The general position argument implies
Proposition 3.1. Two immersions are homotopy equivalent if and only if one immersion can be obtained from the other by a sequence of Reidemeister moves, see Fig. 3.19.
Fig. 3.19Reidemeister moves for circle immersions in a two dimensional surface
Proposition 3.1 can be thought of as definition of the set
In the present section we consider a combinatorial algorithm of free loop recognition, following mainly the article [Hass and Scott, 1994] of J. Hass and P. Scott. Their paper gives a simple answer to the question due to V. G. Turaev [Turaev, 1989]:
Let s0 and s1 be homotopical curves in a surface and each of them have k double points. Does there exist a homotopy st between s0 and s1 such that any curve st has no more than k double points?
The theorems given below are proved in [Hass and Scott, 1994]:
Theorem 3.3. Let s0 and s1 be homotopic curves in general position in a surface and each of them minimise the number of self-crossings
