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Plumbers' knots and unstable Vassiliev theory
by
Chad Giusti
University of Oregon
We begin by constructing the spaces of plumbers' knots, which are piecewise linear with all sticks parallel to the axes. These knots are closely related to lattice knots and provide a new version of finite complexity knot theory which gives rise to classical knot theory. The combinatorial structure of these spaces allows us to construct an algorithm to enumerate knot types with a bounded number of moves. An implementation of this algorithm has demonstrated that, for example, there are seven components of the space of plumbers' knots with five moves, though they all fall into one of three topological types.
We next extend the notion of Vassiliev derivatives to singular knots with singularities other than collections of isolated double-points. We then import Vassiliev's original techniques to the plumber's knots setting, constructing an unstable Vassiliev spectral sequence which is compatible with stabilization (subdivision of segments). This result opens the door to constructing new Vassiliev-style knot invariants and/or seeing the strength of finite-type invariants once we understand the behavior of Vassiliev derivatives under stabilization.
Date received: October 21, 2008
Copyright © 2008 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 # caxq-06.