Untangling trigonal diagrams

Erwan Brugallé (1) , Pierre-Vincent Koseleff (2, 3, 4) , Daniel Pecker (2, 3)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 UPMC - Université Pierre et Marie Curie - Paris 6
3 IMJ - Institut de Mathématiques de Jussieu
4 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs

Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn>0$, 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.

Mots clés: polynomial knots; Two-bridge knots

Domaines: Topologie géométrique [math.GT]; Mathématiques [math]

Erwan Brugallé, Pierre-Vincent Koseleff, Daniel Pecker. Untangling trigonal diagrams. Journal of Knot Theory and Its Ramifications, 2016, 25 (7), ⟨10.1142/S0218216516500437⟩. ⟨hal-01084463v2⟩