Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 22

Below, we shall give a proof of the completeness of the invariant for the braid group elements invented by Artin, see [Gaifullin and Manturov, 2002].

The invariant to be constructed has a simple algebraic description as a map (non-homeomorphic) from the braid group Br(n) to the n copies of the free group in n generators.

Several generalisations of this invariant, such as the spherical and cylindrical braid group invariants, are also complete. The key point of such a completeness is that these invariants originate from several curves, and the braid can be uniquely restored from these curves.

Moreover, this approach finally led to the algorithmic recognition of virtual braids due to Oleg Chterental [Chterental, 2015].

Fig. 2.5Decomposing a pure braid