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Alexander and Markov type theorems for link cobordisms
by
Mark Hughes
SUNY Stony Brook
For surfaces in 4-space we can generalize the notion of classical braid closures to define surface braid closures, which give closed surfaces in R4. Viro proved that any isotopy class of surface link in R4 can be realized as a surface braid closure. Furthermore Kamada gave a set of moves analogous to Markov moves which are sufficient to join any two isotopic surface braid closures through a sequence of surface braid closures. In this talk I will discuss a relative version of these ideas which involves braiding cobordisms in R3 ×[0, 1]. These braided link cobordisms admit particularly nice motion picture descriptions, where all but a finite number of stills are braid closures, while the singular stills each contain a single critical point. We will show that any cobordism in R3 ×[0, 1] between braid closures in R3 ×{0} and R3 ×{1} is isotopic relative its boundary to a braided link cobordism, and that any two braided link cobordisms which are isotopic in R3 ×[0, 1] can be joined by a sequence of Markov type moves. These constructions make use of Morton's diagram threading technique and Kamada's generalization of Markov's theorem for closed surface braids.
Date received: November 20, 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-31.