RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Sb., 2017, Volume 208, Number 6, Pages 55–108 (Mi msb8717)  

This article is cited in 2 scientific papers (total in 2 papers)

Rectangular diagrams of surfaces: representability

I. A. Dynnikov, M. V. Prasolov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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.

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

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This research was funded by a grant from the Russian Science Foundation (project no. 14-50-00005).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8717

Full text: PDF file (1750 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2017, 208:6, 791–841

Bibliographic databases:

ArXiv: 1606.03497
UDC: 515.162.8
MSC: 57M20, 57M25
Received: 12.04.2016 and 14.03.2017

Citation: I. A. Dynnikov, M. V. Prasolov, “Rectangular diagrams of surfaces: representability”, Mat. Sb., 208:6 (2017), 55–108; Sb. Math., 208:6 (2017), 791–841

Citation in format AMSBIB
\Bibitem{DynPra17}
\by I.~A.~Dynnikov, M.~V.~Prasolov
\paper Rectangular diagrams of surfaces: representability
\jour Mat. Sb.
\yr 2017
\vol 208
\issue 6
\pages 55--108
\mathnet{http://mi.mathnet.ru/msb8717}
\crossref{https://doi.org/10.4213/sm8717}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3659579}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2017SbMat.208..791D}
\elib{https://elibrary.ru/item.asp?id=29255289}
\transl
\jour Sb. Math.
\yr 2017
\vol 208
\issue 6
\pages 791--841
\crossref{https://doi.org/10.1070/SM8717}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000408176700003}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85028017619}


Linking options:
  • http://mi.mathnet.ru/eng/msb8717
  • https://doi.org/10.4213/sm8717
  • http://mi.mathnet.ru/eng/msb/v208/i6/p55

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. A. Dynnikov, V. A. Shastin, “On equivalence of Legendrian knots”, Russian Math. Surveys, 73:6 (2018), 1125–1127  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. I. Dynnikov, M. Prasolov, “Classification of Legendrian knots of topological type $7_6$ with maximal Thurston-Bennequin number”, J. Knot Theory Ramifications, 28:14 (2019), 1950089, 13 pp.  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
    Number of views:
    This page:317
    Full text:10
    References:38
    First page:26

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020