Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov
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
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
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