We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S3. These invariants can be used to give homomorphisms from the smooth concordance ...group C to Z, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
Considering a closed monotone Lagrangian submanifold L, we give, under some hypotheses, a lower bound on the intersection number of L with its image by a generic Hamiltonian isotopy. First for ...monotone Lagrangian submanifolds L which are \mathbf {K}(\pi ,1) and, in particular, for monotone Lagrangian submanifolds with negative sectional curvature this bound is 1+\beta _{1}(L). In more general cases the lower bound is weaker. We generalise some results previously obtained by L. Buhovsky in J. Topol. Anal. 2 (2010), pp. 57–75 and P. Biran and O. Cornea in Geom. Topol. 13 (2009), pp. 2881–2989.
We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The ...resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology.In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.
The
ν
+
-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes
C
F
K
∞
, and many concordance ...invariants derived from Heegaard Floer theory are invariant under the relation. In this paper, we show that any genus one knot is
ν
+
-equivalent to one of the trefoil, its mirror and the unknot.
We extend Perutz’s Lagrangian matching invariants to 3-manifolds which are not necessarily fibered using the technology of holomorphic quilts. We prove an isomorphism of these invariants with ...Ozsváth–Szabó’s Heegaard–Floer invariants for certain extremal spinc structures. As applications, we give new calculations of Heegaard–Floer homology of certain classes of 3-manifolds, and a characterization of Juhász’s sutured Floer homology.
Let CT be the subgroup of the smooth knot concordance group generated by topologically slice knots and let CΔ be the subgroup generated by knots with trivial Alexander polynomial. We prove that CT/CΔ ...is infinitely generated. Our methods reveal a similar structure in the 3-dimensional rational spin bordism group, and lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.
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