53 research outputs found
Heegaard Floer homology and integer surgeries on links
Let L be a link in an integral homology three-sphere. We give a description
of the Heegaard Floer homology of integral surgeries on L in terms of some data
associated to L, which we call a complete system of hyperboxes for L. Roughly,
a complete systems of hyperboxes consists of chain complexes for (some versions
of) the link Floer homology of L and all its sublinks, together with several
chain maps between these complexes. Further, we introduce a way of presenting
closed four-manifolds with b_2^+ > 1 by four-colored framed links in the
three-sphere. Given a link presentation of this kind for a four-manifold X, we
then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete
system of hyperboxes for the link. Finally, we explain how a grid diagram
produces a particular complete system of hyperboxes for the corresponding link.Comment: 231 pages, 54 figures; major revision: we now work with one U
variable for each w basepoint, rather than one per link component; we also
added Section 4, with an overview of the main resul
Contact surgeries and the transverse invariant in knot Floer homology
We study naturality properties of the transverse invariant in knot Floer
homology under contact (+1)-surgery. This can be used as a calculational tool
for the transverse invariant. As a consequence, we show that the
Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and
n>3.Comment: Corrected naturality discussion
An overview of knot Floer homology
Knot Floer homology is an invariant for knots discovered by the authors and,
independently, Jacob Rasmussen. The discovery of this invariant grew naturally
out of studying how a certain three-manifold invariant, Heegaard Floer
homology, changes as the three-manifold undergoes Dehn surgery along a knot.
Since its original definition, thanks to the contributions of many researchers,
knot Floer homology has emerged as a useful tool for studying knots in its own
right. We give here a few selected highlights of this theory, and then move on
to some new algebraic developments in the computation of knot Floer homology
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