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Organizers |
Knots, categories, and dynamics
by
Maciej Niebrzydowski
University of Louisiana at Lafayette
The subject of my talk is a generalization of theories involving diagrams and moves on the diagrams, tying them closely with the notions of category theory and transition systems. In a nutshell: we consider diagrams decorated by structures (objects of a given category), the structure determines which moves are possible on a diagram, and the moves change the structure in a controlled way. This creates dynamics. The following is our main example. Given a classical link diagram, suppose that the components are ordered in some way, and that the components with higher order are allowed to move over components with the lower order, but not the other way around. What is the set of diagrams one can obtain in such case? More generally, we can have a binary relation R on the set of arcs of the diagram, and impose the condition that an arc b can move over an arc a only if aR b. We will investigate the consequences of such a condition. In particular, it is necessary to make a choice regarding the status (with respect to R) of the arcs created by the Reidemeister moves. There are some options, and they lead to different kinds of links; the choice can be influenced by the desired applications. In some cases, there arises a phenomenon of irreversible Reidemeister moves which leads to a different definition of a knot (now viewed as a subcategory) and to replacing invariants by indicators (functors). Still, if we decide to use the binary relation in which all the arcs are related, we return to classical knot theory. We define homologies of binary relations and merge them with quandle homology to get a tool for answering some of the questions that arise.
Date received: October 10, 2012
Copyright © 2012 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbfw-04.