Abstracts

Foundational and philosophical problems of pure and applied mathematics

In recent years braid groups are intensively studied, in particular, there is well-known result by Thurston, that these Groups admit biautomatic structure, that yields to normal forms in braids. Unfortunately, these normal forms are very dull from computational point of view and are inconvenient to use in applications.

Recall, that Artin braid group on $n+1$ strands is a group given with the following generators and relations: begin{equation} label{eq:B} B_{n+1}= left. end{equation}

We find new normal forms for braids, which possess extremely natural geometric description.

The main geometric idea of our normal form is to pull down-right the third strand, lowering it till possible. This process allows us to have all crossings that involve first and second strand in the very left upper corner. And the the crossings that involve strands with bigger number lower.

We introduce a new order on $B_{n+1}$ and Knuth-Bendix (or Grobner-Shirshov) type algorithm, that by given braid word calculates its normal form, which turns out to be the least in the sense of our order.

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