Arc-presentations of links: Monotonic simplification

Volume 190 / 2006

I. A. Dynnikov Fundamenta Mathematicae 190 (2006), 29-76 MSC: Primary 57M25. DOI: 10.4064/fm190-0-3


In the early 90's J. Birman and W. Menasco worked out a nice technique for studying links presented in the form of a closed braid. The technique is based on certain foliated surfaces and uses tricks similar to those that were introduced earlier by D. Bennequin. A few years later P. Cromwell adapted Birman–Menasco's method for studying so-called arc-presentations of links and established some of their basic properties. Here we further develop that technique and the theory of arc-presentations, and prove that any arc-presentation of the unknot admits a (non-strictly) monotonic simplification by elementary moves; this yields a simple algorithm for recognizing the unknot. We also show that the problem of recognizing split links and that of factorizing a composite link can be solved in a similar manner. We also define two easily checked sufficient conditions for knottedness.


  • I. A. DynnikovDepartment of Mechanics and Mathematics
    Moscow State University
    Moscow 119992 GSP-2, Russia

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