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Suppose I have the $s_i$ and $s^{-1}_i$ as generators, satisfying the braid relations. I call the $s_i$ "right twists" and their inverses "left twists".

Any element $w$ in the braid group can be written as a product of generators (but not in a unique way). Suppose this representation is reduced, meaning that the number of elements in this product is minimized. Let $rt(w)$ be the maximal exponent of an element $s_i$ in all possible products representing $w$. Think of this as the maximal right-twisting of two strands. Similarly, $lt(w)$ is the maximal exponent of an $s^{-1}_i$.

Now, let $R$ be a product of right twists, meaning it is a product of entries in $\{s_i\}$ but not inverses of these, and let $L$ be a product of left twists. Now let $RL$ be the product of $L$ and $R$.

I want to prove that $rt(R) \geq rt(RL)$ or equivalently, $lt(L) \geq lt(RL)$.

In other words, adding left twists to a braid cannot increase the maximal right twist.

A reference would be superb! This is not a homework problem, but for my research but it seems so simple that there should be an easy proof.

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  • $\begingroup$ Did you mean the last inequality to be $lt(L)\geq lt(RL)$? $\endgroup$
    – Tobias Kildetoft
    Mar 11, 2016 at 20:32
  • $\begingroup$ Hmm, this seems like it should be fairly straightforward by induction on the length of one of them. $\endgroup$
    – Tobias Kildetoft
    Mar 11, 2016 at 20:34
  • $\begingroup$ @TobiasKildetoft: Yes, that's what the next sentence says: its enough to show that a left-twist does not increase the maximal right-twist number.. $\endgroup$
    – Per Alexandersson
    Mar 11, 2016 at 20:35
  • $\begingroup$ So it seems like this reduces to something of the form $ws_{i+1}s_i$ multiplied by $s_{i+1}^{-1}$. $\endgroup$
    – Tobias Kildetoft
    Mar 11, 2016 at 20:37

1 Answer 1

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I figured out an answer, I cross-posted on MO due to the lack of interest here.

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  • $\begingroup$ I don't understand your answer. For example, for $R=s_1^2 s_2^2 s_1$ we have $C(1,2)=3$, but I think that $rt(R)=2$. $\endgroup$
    – san
    Mar 20, 2016 at 4:42

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