Arc-presentations of links: Monotonic simplification

I. A. Dynnikov. "Arc-presentations of links: Monotonic simplification." Fundamenta Mathematicae 190.1 (2006): 29-76. <http://eudml.org/doc/283163>.

@article{I2006,
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.},
author = {I. A. Dynnikov},
journal = {Fundamenta Mathematicae},
keywords = {knot; link; arc-presentation; rectangular diagram; closed braid; foliation},
language = {eng},
number = {1},
pages = {29-76},
title = {Arc-presentations of links: Monotonic simplification},
url = {http://eudml.org/doc/283163},
volume = {190},
year = {2006},
}

TY - JOUR
AU - I. A. Dynnikov
TI - Arc-presentations of links: Monotonic simplification
JO - Fundamenta Mathematicae
PY - 2006
VL - 190
IS - 1
SP - 29
EP - 76
AB - 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.
LA - eng
KW - knot; link; arc-presentation; rectangular diagram; closed braid; foliation
UR - http://eudml.org/doc/283163
ER -