96 research outputs found
Knots which admit a surgery with simple knot Floer homology groups
We show that if a positive integral surgery on a knot K inside a homology
sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y
which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the
standard sphere, the three-manifold Y is a L-space and the Heegaard Floer
homology groups of K are determined by its Alexander polynomial
Filtration of Heegaard Floer homology and gluing formulas
We introduce an extra filtration of \CFK(Y,K) and use it in order to obtain
formulas for Floer homology of , which is obtained from by gluing the knot complements on the framed torus boundaries
Correction to the article: Floer homology and splicing knot complements
This note corrects the mistakes in the splicing formulas of the paper "Floer
homology and splicing knot complements". The mistakes are the result of the
incorrect assumption that for a knot inside a homology sphere , the
involution on the knot Floer homology of which corresponds to moving the
basepoints by one full twist around is trivial. The correction implicitly
involves considering the contribution from this (possibly non-trivial)
involution in a number of places
Seifert fibered homology spheres with trivial Heegaard Floer homology
We show that among Seifert fibered integer homology spheres, Poincare sphere
(with either orientation) is the only non-trivial example which has trivial
Heegaard Floer homology. Together with an earlier result, this shows that if an
integer homology sphere has trivial Heegaard Floer homology, then it is a
connected sum of a number of Poincare spheres and hyperbolic homology spheres
Embedded curves and Gromov-Witten invariants of three-folds
Associated with a prime homology class (i.e.
and imply or is an odd prime)
on a symplectic three-manifold with vanishing first Chern class, we count the
embedded perturbed pseudo-holomorphic curves in of a fixed genus to
obtain certain integer valued invariants analogous to Gromov-Witten invariants
of
Heegaard Floer homologies of pretzel knots
We compute the Ozsv\'ath-Szab\'o Heegaard Floer homology of three stranded
pretzel knots
Floer Cohomology of Certain pseudo-Anosov Maps
Floer cohomology is computed for certain elements of the mapping class group
of a surface of genus which are compositions of positive and
negative dehn twists along some loops in . The computations cover a
certain class of pseudo-Anasov maps.Comment: 19 pages, 4 figure
Knots in homology spheres which have simple knot Floer homology are trivial
We show that if K is a non-trivial knot inside a homology sphere X, the rank
of the knot Floer homology group associated with K is strictly bigger than the
rank of the Heegaard Floer homology group associated with X.Comment: This paper has been withdrawn by the author. The surgery formulas are
used incorrectly for obtaining the main result of the pape
Bordered Floer homology and existence of incompressible tori in homology spheres
Let denote a knot inside the homology sphere . The zero-framed
longitude of gives the complement of in the structure of a bordered
three-manifold, which may be denoted by . We compute the
quasi-isomorphism type of the bordered Floer complex of in terms of the
knot Floer complex associated with . As a corollary, we show that if a
homology sphere has the same Heegaard Floer homology as it does not
contain any incompressible tori. Consequently, if is an irreducible
homology sphere -space then is either , or the Poicar\'e sphere
, or it is hyperbolic
On finiteness and rigidity of J-holomorphic curves in symplectic three-folds
Given a symplectic three-fold we show that for a generic almost
complex structure which is compatible with , there are finitely
many -holomorphic curves in of any genus representing a
homology class in \H_2(M,\Z) with , provided that the
divisibility of is at most 4 (i.e. if with and then ). Moreover, each such curve is embedded
and 4-rigid.Comment: This is a revision of the original submission. The assumption on the
homology class is imposed in order to fill the gap in the original versio
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