Non-Reidemeister Knot Theory and Its Applications in Dynamical Systems, Geometry, and Topology

Abstract

Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free $k$-braid group}. In the simplest case, for free $2$-braids, the word problem and the conjugacy problem can be solved by finding the minimal representative, which can be thought of as a graph, and is unique, as such. We prove a general theorem about invariants of dynamical systems which are valued in such groups and hence, in pictures. We describe various applications of the above theory: invariants of weavings (collections of skew lines in $\R^{3}$), and many other objects in geometry and topology. In general, provided that for some topological objects (considered up to isotopy, homotopy etc) some easy axioms (coming from some dimensional constraints) hold, one can construct similar dynamical systems and picture-valued invariants.