Miscellaneous papers

• [1] 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.

• [2] An introduction to the Seiberg-Witten equations on symplectic four-manifolds (with C. H. Taubes),
  Symplectic geometry and topology (Park City, UT, 1997), 103-142, AMS, 1999. pdf (may differ slightly from published version)
  This expository article is based on Taubes's lectures at Park City, Utah in July 1997. Lecture 1: Background from differential geometry. Lecture 2: Spin and the Seiberg-Witten equations. Lecture 3: The Seiberg-Witten invariants. Lecture 4: The symplectic case, part I. Lecture 5: The symplectic case, part II.

• [3] Topological bifurcations of attracting 2-tori of quasiperiodically driven oscillators (with B. Spears and A. Szeri),
  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.)

• [4] Floer homology of families I
  Algebraic and Geometric Topology 8 (2008), 435-492. download.
  Abstract: In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. This filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory, some analysis needs to be carried out in order to turn this principle into a theorem. This paper constructs the spectral sequence in detail for the model case of finite-dimensional Morse homology, and shows that it recovers the Leray-Serre spectral sequence of a smooth fiber bundle. We also generalize from Morse homology to Novikov homology, which involves some additional subtleties.

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