Giroux torsion is an important class of contact structures on a neighborhood of a torus, which is known to obstruct symplectic fillability. Ghiggini conjectured that half Giroux torsion along a separating torus always results in a vanishing Heegaard Floer contact invariant hence also obstructs fillability. In this talk, we present a counterexample to that conjecture. Our main tool is a bordered contact invariant, which enables efficient computation of the contact invariant.
Satellite operations are one of the most basic operations in knot theory. Many researchers have studied the behavior of knot Floer homology under satellite operations. Most of these results use Lipshitz, Ozsvath and Thurston's bordered Heegaard Floer theory. In this talk, we discuss a new technique for studying these operators, and we apply this technique to a family of operators called L-space operators. Using this theory, we are able in many cases to give a simple formula for the behavior of the concordance invariant tau under such operators.
It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.
Giroux and Pardon conjectured that a Lagrangian L in a Weinstein manifold W is regular (that is, compatible with the Weinstein structure in a natural sense) if there is a Lefschetz fibration p: W \to \C such that p(L) is a ray. In this talk, I will discuss forthcoming joint work with A. Roy and L. Wang, which establishes this conjecture. As an application of the proof, we show how all fillings of the rainbow closures of a positive braid can be described by manipulations of arcs in the base of an appropriate Lefschetz fibration.
Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.
This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift).
We discuss the proof of the following Theorem
Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.
The Jones polynomial was first defined by Vaughan Jones as a "trace function" on an algebra discovered via operator algebras. It was discovered that the polynomial satisfies certain skein relations. The HOMFLY polynomial was discovered through both skein relations and a "lift" of the trace function on the Jones algebra to the "Hecke algebra". Another 2-variable polynomial called the Kauffman polynomial was discovered purely via skein relations. In this talk, we discuss how the process started by Jones was reversed for this polynomial.
I'll talk about some specialized kirby calculus constructions: immersed surface complements and round handles. I'll prove using kirby calculus that S2xS2 minus an appropriate smooth embedded S2vS2 is diffeomorphic to R4. Maybe that is obvious, but the point is we can find nice diagrams where you see everything explicitly.
