Untangling trigonal diagrams
  Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases. 
Erwan Brugall\'e (CMLS-EcolePolytechnique), Pierre-Vincent Koseleff   (UPMC, IMJ, INRIA Paris-Rocquencourt), Daniel Pecker (...
math.GT