Integration of singular braid invariants and graph cohomology,
Transactions of the AMS 350 (1998), 1791-1809. Postscript PDF
This paper attempts to explain some of the mysteries underlying the
existence of Vassiliev knot invariants from a topological point of
view, in the easier case of braids. In particular we prove necessary
and sufficient conditions for an arbitrary invariant of singular
braids with m double points to be ``integrable'' to a braid invariant.
This gives a slight generalization of the existence theorem for
Vassiliev invariants of braids.
 Topological bifurcations of attracting 2-tori of
quasiperiodically driven oscillators (with B. Spears and
Journal of Nonlinear Science 15 (2005), 423-452.
I was a "topology consultant" for this paper, which studies some
knotted 2-tori in R^2 x T^2 that arise as attractors in a certain
dynamical system of interest in mechanical engineering. Along the
way, I was surprised to learn that the following seemingly basic
topological question is apparently unsolved: if two closed braids are
isotopic as links in R^2 x S^1, then must they be isotopic through
closed braids? (For braids in R^2 x I, the answer is yes by a
classical theorem of Artin.)
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