The rotating normal form of braids is regular

Abstract

Defined on Birman-Ko-Lee monoids, the rotating normal form has strong connections with the Dehornoy's braid ordering. It can be seen as a process for selecting between all the representative words of a Birman-Ko-Lee braid a particular one, called rotating word. In this paper we construct, for all n 2, a finite-state automaton which recognizes rotating words on n strands, proving that the rotating normal form is regular. As a consequence we obtain the regularity of a $σ$-definite normal form defined on the whole braid group.

Figures (9)

• Figure 1: Interpretation of a word in the letters $\sigma_{\!i}^{\space\hbox{$\pm$}\space1}$ as a geometric braid diagram.
• Figure 2: In the geometric braid $a_{1,4}$, the strands $1$ and $4$ cross under the strands $2$ and $3$.
• Figure 3: Rolling up the usual braid diagram helps us to visualize the symmetries of the braids $a_{p,q}$. On the resulting cylinder, $a_{p,q}$ naturally corresponds to the chord connecting vertices $p$ and $q$. With this representation, $\phi_{\space\hbox{$n$}}$ acts as a clockwise rotation of the marked circles by $2\pi/n$.
• Figure 4: The $\phi_{\space\hbox{$6$}}$-splitting of a braid of $B_{6}^{\space\hbox{$+$}\space*}$. Starting from the right, we extract the maximal right-divisor that keeps the sixth strand unbraided, then extract the maximal right-divisor that keeps the first strand unbraided, etc.
• Figure 5: An $a_{2,6}$-ladder. The gray line starts at position $1$ and goes up to position $5$ using the bar of the ladder. The empty spaces between bars in the ladder are represented by a framed box. In such boxes the vertical line representing the letter $a_{i,j}$ does not cross the gray line. The bar of the ladder are represented by black thick vertical lines.

Theorems & Definitions (51)

• Proposition 1.1
• Definition 2.1
• Proposition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Definition 2.7
• Lemma 2.8: Lemma 3.2 of Fromentin2008a
• Definition 2.9