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

      Remark 2.4. If we put t = 1, the results f(X) and f(Y) become the same. Thus that is the variable t that “feels” the forbidden move.

      2.4.2.1A 2n-variable generalisation of the invariant

      In the present section we define a stronger version of the f invariant described in the previous section. This generalised invariant was invented by V. O. Manturov soon after the paper [Chterental, 2015] was published; however, the definition remained unpublished since the invariant of (n + 1) variables itself was conjecturally complete.

      We begin with the group — the free group in generators a₁, . . . , t₁, . . . , tn and denote by the quotient sets of right residual classes {aj}\G for i = 1, . . . , n.

      Definition 2.18. An extended virtual n-system is a set of elements .

      Now we construct an invariant which takes values in extended virtual n-systems. The construction follows the same pattern as in the case of the f invariant, but with a different approach to virtual crossings. We begin with a system e, . . . , e and process the crossings one by one.

      To be precise, consider a crossing corresponding to an i-th generator or its inverse: either σi, ζi or . Assume that the left strand of this crossing originates from the point (p, 1), and the right one originates from the point (q, 1). Let ep = 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 . If the letter is , then eq stays the same, and ep becomes PQ⁻¹aqQ. Finally, if the letter is ζi, then ep becomes P · tq, and eq becomes . This operation is well defined.

      Obviously, the function collapses to the function f defined in the previous section if we “forget” the distinction between the variables t₁, . . . , tn.

      Along the same lines as in the previous section the following theorem is verified:

      Theorem 2.8. The function is an invariant of virtual braids.

      Unlike the case of invariant f, though, the following conjecture remains open.

      Conjecture 2.1. The invariant is complete.

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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

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.

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