Arc-presentations of links: Monotonic simplification

Abstract

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.