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
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 →
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
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
