# Braid group: Can a left-twist increase the number of right twists?

Disclaimer: This question was first posted on math.se without any answer.

This is something that naturally occurs in my research, but I am no expert on this - it feels like a natural question so I am hoping for a reference or a short proof.

Consider braids on $n$ strands with the group generated by $s_1,\dotsc, s_{n-1}$, left-twists, and their inverses, right twists.

Remember the classical braid relations, $s_i s_j = s_j s_i$ if $|i-j| \geq 2$, and $s_i s_{i+1}s_i=s_{i+1} s_{i}s_{i+1}$.

Let $R(w)$ be all reduced words for an element $w$ in the braid group, and let $RT(w)$ be defined as the maximal positive exponent that appears, among all representatives of $w$ in $R(w)$. That is, $RT(w)$ somehow captures the maximal number of consecutive right twists among all representatives of $w$, without cheating by introducing more right twists followed by same number of left twists (remember, we only consider reduced words).

Q1: I would like to prove that $RT(uw) \leq RT(u)+RT(w)$.

Q2: The special case that I really care about arises from permutations: let $u$ and $w$ be reduced words of permutations in $S_n$, meaning no $s_i^2$ appears.
Re-interpret these two words as elements in the braid group (by doing right-twists) and consider the braid $b = uw^{-1}$, a bunch of right-twists followed by left-twists.

Is it true that for any reduced representative of $b$ in the braid group, there is no exponent greater than $1$, i.e., $s_i^2$ is not a subword?

In English - can left-twists increase the number of right-twists?

Note: Of course $uw^{-1}$ might not be reduced, one can use braid relations to cancel some left-twists with right twists.

## 1 Answer

I think I figured out what to do:

One introduce $C_{ij}(w)$ as the (signed) number of times strand $i$ crosses above $j$, and consider the maximum and minimum of all pairs of strands. This basically gives the same as the RT statistic.

Now, using THIS definition, it is easy to follow pairs of strands under braid composition, which implies the statements above.