2017/12/31 by Ivan Dynnikov, Maxim Prasolov · 5 citations
Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Class (philosophy) #Diagram #Geometric and Algebraic Topology #Isotopy #Quasicrystal Structures and Properties #Regular polygon #Surface (topology) #Type (biology) #math.GT #msc:57M25 #msc:57M50 #msc:57R17
paper · pdf · doi:10.1112/topo.12194
published in Journal of Topology 14(3), 701-860 (Wiley) · 146 pages, 110 figures. This is a close-to-final version of the paper, to appear in Journal of Topology
openalex created_date 2018/01/05 · arxiv created 2021/03/29 · openalex publication_date 2021/07/02 · arxiv updated 2021/07/20 · openalex updated_date 2026/08/05
In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce basic moves that allow the transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves are divided into two (non-exclusive) types so that, vaguely speaking, type I moves commute with type II ones. This commutation is the matter of the main technical result of the paper. We use it as well as a relation of the moves to Giroux's convex surfaces to propose a new method for distinguishing Legendrian knots. We apply this method to show that two Legendrian knots having topological type 6 2 are not equivalent. More applications of the method will be the subject of subsequent papers. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.