38 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
Branched covers of quasipositive links and L-spaces
Let be a oriented link such that , the -fold cyclic cover
of branched over , is an L-space for some . We show that if
either is a strongly quasipositive link other than one with Alexander
polynomial a multiple of , or is a quasipositive
link other than one with Alexander polynomial divisible by , then there is an integer , determined by the Alexander
polynomial of in the first case and the Alexander polynomial of and the
smooth -genus of , , in the second, such that . If
is a strongly quasipositive knot with monic Alexander polynomial such as an
L-space knot, we show that is not an L-space for , and
that the Alexander polynomial of is a non-trivial product of cyclotomic
polynomials if is an L-space for some . Our
results allow us to calculate the smooth and topological 4-ball genera of, for
instance, quasi-alternating quasipositive links. They also allow us to classify
strongly quasipositive alternating links and -strand pretzel links.Comment: 49 pages, 7 figures, minor corrections and improved exposition,
accepted for publication by the Journal of Topolog
Reducible And Finite Dehn Fillings
We show that the distance between a finite filling slope and a reducible
filling slope on the boundary of a hyperbolic knot manifold is at most one.Comment: 17 pages, 11 figure
On definite strongly quasipositive links and L-space branched covers
We investigate the problem of characterising the family of strongly
quasipositive links which have definite symmetrised Seifert forms and apply our
results to the problem of determining when such a link can have an L-space
cyclic branched cover. In particular, we show that if is the dual Garside element and is a strongly quasipositive braid whose braid closure is
definite, then implies that is one of the torus links
or pretzel links . Applying
Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic
branched covers of is an L-space, then is one of
these links. We show by example that there are strongly quasipositive braids
whose closures are definite but not one of these torus or pretzel
links. We also determine the family of definite strongly quasipositive
-braids and show that their closures coincide with the family of strongly
quasipositive -braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat