Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 28

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was invented by V. O. Manturov soon after the paper [Chterental, 2015] was published; however, the definition remained unpublished since the invariant of (n + 1) variables itself was conjecturally complete.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].