Instructions: click on blue and green circles until the number of crossings in the picture is minimal possible (see 'Minimal possible length' below).
Both types of circles correspond to conjugation moves $f\mapsto s_i f s_i$, but blue circles preserve the number $\ell(f)$ of crossings, and green circles decrease it by 2.
Press the "Run ODE" button to solve the problem using a system of linear ODE: $$ x_i(0) = i;\quad x'_i(t) = x_{f(i)}-x_i \quad \forall i\in\mathbb{Z},$$
where $x_i(t)$ is the horizontal coordiante of the two points labeled $i$ at the top and bottom row of the picture. See the proof of Proposition 4.13 in arXiv:2212.12962 for more details.
$[f(1),f(2),...,f(n)]=$
Speed:
Length: $\ell(f) = $ . Minimal possible length: .
$k(f) = $ , $n(f) = $ . Number of cycles: $\operatorname{ncyc}(f) = $ .
and green 
circles until the number of crossings in the picture is minimal possible (see 'Minimal possible length' below). 