Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 29

We start with basic definitions of knot theory [Manturov, 2018].

A classical knot (a classical link) is a smooth embedding of the circle S¹ (a disjoint union of circles to three dimensional space ³ (or three dimensional sphere S³). Knots and links are usually considered up to isotopies in ³.

The natural orientation of the circle S¹ induces an orientation of the knot (link).

A conventional way to present knots and links is based on their plane generic projections — link diagrams.

A classical link diagram is a 4-valent plane graph, each vertex of which is endowed with undercrossing-overcrossing structure, see Fig. 3.1. The graph can have also circle components without crossings.

Fig. 3.1Knot crossing

Isotopic links may give different diagrams after projection, but this freedom is controlled by Reidemeister’s theorem [Reidemeister, 1948].

Theorem 3.1. Two link diagrams D₁ and D₂ correspond to the same link isotopy class if and only if the diagram D₂ can be obtained from D₁ by a sequence of diagram transformations, called Reidemeister moves, see Fig. 3.2, and diagram isotopies.

Fig. 3.2Reidemeister moves

Thus, one can define links (and knots) as equivalence classes of link diagrams modulo Reidemeister moves and diagram isotopies.

Link diagrams do not carry natural orientations so their equivalence classes determine nonoriented links. In order to define an oriented link, one should orient all the edges of a link diagram so that opposite edges of any crossing of the diagram have the same orientation. Such an orientation is compatible with Reidemeister moves and the corresponding equivalence class of oriented diagrams determines an oriented link.

Diagrams of the simplest knots and links are given in Fig. 3.3

Fig. 3.3Simplest knots and links

The main question of knot theory is the knot recognition problem: which two knots are (isotopic) and which are not? A partial case of the knot recognition problem is the trivial knot recognition problem. Here, trivial knot (or unknot) means the simplest knot that can be represented as the boundary of a 2-disc embedded in ³. Both questions are very difficult.

As usual, in order to prove that two knot diagrams correspond to the same knot, one should present a sequence of Reidemeister moves which transforms the first diagram to the second one. The difficulty is that the intermediate diagrams can be much more complicated than the initial ones. For example, the diagram of the unknot in Fig. 3.4 cannot be reduced by Reidemeister moves to the trivial diagram (Fig. 3.3, 1) without adding new crossings to the diagram.

Fig. 3.4A diagram of the unknot

In order to prove two knot diagrams are not equivalent, one should construct a knot invariant that distinguish these diagrams. A knot (link) invariant is a function on the representatives of knots and links (embeddings, diagrams etc.) whose value does not change if one replaces a representative of a knot (link) with another representative of the same knot (link). So if an invariant has different values on two diagrams, then the corresponding knots (links) are different.

One of the most famous and useful knot invariants is Jones polynomial, see for example [Manturov, 2018].

Given a (nonoriented) link diagram D with the set of crossings χ(D), consider the set of states S(D) = {0, 1}^χ(D). For each state s ∈ S(D) we can define a diagram Ds which appears from D by smoothing of the diagram according the state s. The rule for smoothing is shown in Fig. 3.5.

Fig. 3.5Types of smoothing

Let α(s) be the number of 0 in s and β(s) be the number of 1 in s. The diagram Ds has no crossings, i.e. it is a union of circles. Let γ(s) be the number of circles in the diagram Ds. The polynomial

is called the Kauffman bracket of the diagram D.

The Kauffman bracket is invariant under second and third Reidemeister moves, but the first Reidemeister move multiplies the Kauffman bracket by −a^±3. A genuine invariant appears after normalising the bracket by an appropriate factor. The normalisation uses a knot orientation. Let

be the writhe number of an oriented link diagram D where the sign of a crossing c is calculated according to Fig. 3.6.

Fig. 3.6Sign of a crossing

The polynomial is called the Jones polynomial of the link diagram D with given orientation. The properties of Jones polynomial can be summarized as follows, see for example [Manturov, 2018].

Theorem 3.2.

(1)Jones polynomial X(D) is an invariant of oriented links;

(2)Jones polynomial obeys the skein relation

The arguments here are any oriented link diagrams which coincide everywhere except a small neighbourhood inside which they look like the corresponding icons.

(3)For any oriented links L₁ and L₂ we have X(L₁#L₂) = X(L₁)X(L₂) where L₁#L₂ is a connected sum of the links, see Fig. 3.7.

Fig. 3.7Connected sum of links

It is yet unknown, whether the Jones polynomial recognises the trivial knot.

Another way to present an oriented knot (not link) is its Gauß diagram (also called a chord diagram). Given a knot diagram D, it can be treated as an immersion S¹ → ². Consider the preimages of the double points and connect any two preimages, corresponding to the same crossing, by an edge, see Fig. 3.8.

Fig. 3.8A Gauß diagram

The edges of the resulting chord diagrams have orientation and signs. The orientation is induced by the undercrossing-overrossing structure; any edge is oriented from the overcrossing to the undercrossing. The edge signs come from the orientation of the immersion and coincides with the signs of the corresponding crossings, see Fig. 3.6.

The knot diagram can be restored from its Gauß diagram up to isotopy (and pass of arcs through the infinite point of ²).

Given a link diagram, one can apply the same construction and obtain a Gauß diagram which will have several oriented circles and chord (with orientations and signs) between them, see Fig. 3.9.

Fig. 3.9Whitehead link and its Gauß diagram

Reidemeister moves on knot diagrams induce moves on Gauß diagrams, see Fig. 3.10.

On the other hand, there are Gauß diagrams which do not correspond to any classical knot diagram, for example see in Fig. 3.11. Any attempt to draw a diagram of the knot in the plane leads to an additional crossing (marked with a circle in the figure). This fact can be proved by the parity argument: any chord in the Gauß diagram of a classical knot can intersect only even number of the other chords, but in the given Gauß diagram the both chords are odd in this sense.

Fig. 3.10Reidemeister moves on Gauß diagrams

Fig. 3.11Nonclassical Gauß diagram

This disparity was one of the motivations to enhance the notion of knots and to introduce virtual knots and links.

One can define a virtual knot as an equivalence class of a Gauß diagram modulo Reidemeister moves on Gauß diagrams.

Another way to define virtual knots (and links) is to consider virtual link diagrams. A virtual link diagram is a 4-valent plane graph, whose vertices are split into two types: classical vertices with undercrossing-overcrossing structure (see Fig. 3.1) and virtual vertices marked with circles (see Fig. 3.12).

Fig. 3.12Virtual crossing

The admissible transformation of virtual diagrams include classical Reidemeister moves (see Fig. 3.2) and virtual Reidemeister moves, see Fig. 3.13.

Local virtual Reidemeister moves can be replaced with one detour move, see Fig. 3.14. A detour move replaces an arc, containing only virtual crossing, with another arc with the same ends that contains only virtual crossings as well.

Fig. 3.13Virtual Reidmeister moves

Fig. 3.14Detour move

Now, we can define a virtual link as an equivalence class of virtual link diagrams modulo classical and virtual Reidemeister moves (or classical Reidemeister moves and the detour move).

The third approach to virtual links employs considering knots and links in thickenings of two dimensional surfaces. Let Σ be an oriented closed two dimensional surface. A link in the thickening of the surface Σ is an embedding a disjoint union of circles 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