Let $B_{n}$ denote the $n$-strand braid group. Let $B_{[m,n]}$ denote the subgroup of $B_{n}$ generated by the strands $\sigma_{m},...,\sigma_{n-1}$. If $b$ is a braid, then let $|b|$ denote the length of the shortest braid word that represents $b$.

Suppose now that $n,r,s$ are natural numbers with $1<s<r<n$. If $b\in B_{n}$ is a braid chosen at random, then what are some bounds for the expected value of $\textrm{Min}\{|abc|:a\in B_{r},c\in B_{[s,n]}\}$. I am especially interested in the cases where $s$ and $n-r$ are small compared to $n$. I am especially interested in the case when $s=2$. I will let the answerer choose the distributions to use. Heuristic and experimental answers are welcome.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.