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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-3,
author = {I. A. Dynnikov},
title = {Arc-presentations of links: Monotonic simplification},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {29-76},
zbl = {1132.57006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-3}
}
I. A. Dynnikov. Arc-presentations of links: Monotonic simplification. Fundamenta Mathematicae, Tome 189 (2006) pp. 29-76. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm190-0-3/