= P, eq = Q, where P, Q are some words representing the corresponding residue classes. After the crossing of all residue classes but ep, eq should stay the same.
Then if the letter is σi, then ep stays the same, and eq becomes
Obviously, the function
Along the same lines as in the previous section the following theorem is verified:
Theorem 2.8. The function
Unlike the case of invariant f, though, the following conjecture remains open.
Conjecture 2.1. The invariant
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1In fact, there are other braid groups called Brieskorn braid groups. For more details see [Brieskorn, 1971; Brieskorn, 1973].
Chapter 3
Curves on Surfaces. Knots and Virtual Knots
3.1Basic notions of knot theory
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
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
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.
