Rectangular diagrams of surfaces: representability

I. A. Dynnikov ; M. V. Prasolov

Sbornik. Mathematics, Tome 208 (2017) no. 6, pp. 791-841

Résumé

Introduced here is a simple combinatorial way, which is called a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular diagrams of links. By using rectangular diagrams of surfaces it is intended, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram? Roughly speaking, the answer is this: there is no restriction on the isotopy class of the surface, but there is a restriction on the rectangular diagram of the boundary link arising from the presentation of the surface. The result extends to Giroux's convex surfaces for which this restriction on the boundary has a natural meaning. In a subsequent paper, transformations of rectangular diagrams of surfaces will be considered and their properties will be studied. By using the formalism of rectangular diagrams of surfaces an annulus in $\mathbb S^3$ is produced here that is expected to be a counterexample to the following conjecture: if two Legendrian knots cobound an annulus and have zero Thurston-Bennequin numbers relative to this annulus, then they are Legendrian isotopic. Bibliography: 30 titles.

Détail

Keywords: rectangular diagram, Legendrian knot, contact structure
Mots-clés : convex surface in Giroux's sense.