7,663 research outputs found
Dehn surgeries on knots in product manifolds
We show that if a surgery on a knot in a product sutured manifold yields the
same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The
proof uses techniques from sutured manifold theory.Comment: 21 pages, 5 figure
Heegaard Floer homology and fibred 3--manifolds
Given a closed 3--manifold , we show that the Heegaard Floer homology
determines whether fibres over the circle with a fibre of negative Euler
characteristic. This is an analogue of an earlier result about knots proved by
Ghiggini and the author.Comment: Version 3: 16 pages, 1 figure. This version incorporates the
corrigendum to a previous paper. To appear in American Journal of
Mathematics. Version 2: Corrects some mistakes in Version 1. The last section
of Version 1 is replaced using a quite different and much simpler method.
Exposition improved according to the referee's suggestion
Heegaard Floer correction terms and rational genus bounds
Given an element in the first homology of a rational homology 3-sphere ,
one can consider the minimal rational genus of all knots in this homology
class. This defines a function on , which was
introduced by Turaev as an analogue of Thurston norm. We will give a lower
bound for this function using the correction terms in Heegaard Floer homology.
As a corollary, we show that Floer simple knots in L-spaces are genus
minimizers in their homology classes, hence answer questions of Turaev and
Rasmussen about genus minimizers in lens spaces.Comment: 21 pages. V2: corrects a mistake in the proof of Proposition 1.5,
incorporates the referee's comment
Characterizing slopes for torus knots
A slope is called a characterizing slope for a given knot in
if whenever the -surgery on a knot in is homeomorphic
to the -surgery on via an orientation preserving homeomorphism,
then . In this paper we try to find characterizing slopes for torus
knots . We show that any slope which is larger than the
number is a characterizing slope for .
The proof uses Heegaard Floer homology and Agol--Lackenby's 6--Theorem. In the
case of , we obtain more specific information about its set of
characterizing slopes by applying more Heegaard Floer homology techniques.Comment: Version 2: 19 pages. This is a major revision. The title of the first
version was "Towards a Dehn surgery characterization of ". We
extended the result in the first version to general torus knots. We also
fixed a gap in the first version, so our result for is slightly
weaker than the originally claimed on
Dehn surgery on knots in producing Nil Seifert fibred spaces
We prove that there are exactly Nil Seifert fibred spaces which can be
obtained by Dehn surgeries on non-trefoil knots in , with as the exact set of all such surgery slopes up to taking the mirror
images of the knots. We conjecture that there are exactly specific
hyperbolic knots in which admit Nil Seifert fibred surgery. We also give
some more general results and a more general conjecture concerning Seifert
fibred surgeries on hyperbolic knots in .Comment: 11 page
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