Topology Atlas | Conferences

Robust bases and transformations of knotted cycles
by
Paul Kainen
Department of Mathematics, Georgetown University

Let G be a graph. An algebraic circuit is a set of edges with the property that every vertex is incident with an even number of edges - i.e., that induces an Eulerian subgraph. A cycle is a connected subgraph that is 2-regular. A circuit basis B is called robust provided that every cycle in the basis is a cycle and for any cycle z in G there exists an ordering (b₁, b₂, ¼, b_r) of a subset B¢ of B such that we have (i) z = b₁ + b₂ + ¼+ b_r and (ii) for every 0 < s < r   , B_s+1 Ç  B₁ + ¼+ B_s is a nontrivial path. It follows that all of the partial sums are cycles.

If G is topologically embedded in 3-dimensional space in such a way that some cycle is knotted, then B must also contain a knotted cycle. We consider some generalizations involving various knot-theoretic invariants. We also study the effect of using a weaker notion of robustness in which the partial sums are still required to be cycles but the intersections between each new cycle and the preceding partial sum can be the union of more than two nontrivial paths. In the case of an overlap along three such paths, two unknotted cycles can give a knotted one, but this does not seem to be possible in the case that the overlap involves only two nontrivial paths.

This suggests a model for rapid topological transformation of knotted cycles involving several "active sites" and, hence, some questions regarding protein actions on polymers arise.

Date received: February 10, 2005

Copyright © 2005 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 # capo-31.