5,876 research outputs found
Combinatorial methods in Dehn surgery
This is an expository paper, in which we give a summary of some of the joint
work of John Luecke and the author on Dehn surgery. We consider the situation
where we have two Dehn fillings and on a given
3-manifold , each containing a surface that is either essential or a
Heegaard surface. We show how a combinatorial analysis of the graphs of
intersection of the two corresponding punctured surfaces in enables one to
find faces of these graphs which give useful topological information about
and , and hence, in certain cases, good upper bounds on
the intersection number of the two filling slopes
On surface subgroups of doubles of free groups
We give several sufficient conditions for a double of a free group along a
cyclic subgroup to contain a surface subgroup.Comment: 21 pages, 1 figur
Knots with small rational genus
If K is a rationally null-homologous knot in a 3-manifold M, the rational
genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces
S in the complement of K whose boundary wraps p times around K for some p
(hereafter: S is a p-Seifert surface for K). Knots with very small rational
genus can be constructed by "generic" Dehn filling, and are therefore extremely
plentiful. In this paper we show that knots with rational genus less than 1/402
are all geometric -- i.e. they may be isotoped into a special form with respect
to the geometric decomposition of M -- and give a complete classification. Our
arguments are a mixture of hyperbolic geometry, combinatorics, and a careful
study of the interaction of small p-Seifert surfaces with essential subsurfaces
in M of non-negative Euler characteristic.Comment: 38 pages, 3 figures; version 3 corrects minor typos; keywords: knots,
rational genu
Taut foliations, left-orderability, and cyclic branched covers
We study the question of when cyclic branched covers of knots admit taut
foliations, have left-orderable fundamental group, and are not L-spaces.Comment: Corrected error in statement and proof of Theorem 1.4 in previous
arXiv and published versions. 34 pages, 13 figures, 2 table
Bridge number and integral Dehn surgery
In a 3-manifold M, let K be a knot and R be an annulus which meets K
transversely. We define the notion of the pair (R,K) being caught by a surface
Q in the exterior of the link given by K and the boundary curves of R. For a
caught pair (R,K), we consider the knot K^n gotten by twisting K n times along
R and give a lower bound on the bridge number of K^n with respect to Heegaard
splittings of M -- as a function of n, the genus of the splitting, and the
catching surface Q. As a result, the bridge number of K^n tends to infinity
with n. In application, we look at a family of knots K^n found by Teragaito
that live in a small Seifert fiber space M and where each K^n admits a Dehn
surgery giving the 3-sphere. We show that the bridge number of K^n with respect
to any genus 2 Heegaard splitting of M tends to infinity with n. This contrasts
with other work of the authors as well as with the conjectured picture for
knots in lens spaces that admit Dehn surgeries giving the 3-sphere
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