39 research outputs found
Satellite operators as group actions on knot concordance
Any knot in a solid torus, called a pattern or satellite operator, acts on
knots in the 3-sphere via the satellite construction. We introduce a
generalization of satellite operators which form a group (unlike traditional
satellite operators), modulo a generalization of concordance. This group has an
action on the set of knots in homology spheres, using which we recover the
recent result of Cochran and the authors that satellite operators with strong
winding number give injective functions on topological concordance
classes of knots, as well as smooth concordance classes of knots modulo the
smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite
operators yields a characterization of surjective satellite operators, as well
as a sufficient condition for a satellite operator to have an inverse. As a
consequence, we are able to construct infinitely many non-trivial satellite
operators P such that there is a satellite operator for which
is concordant to K (topologically as well as smoothly in a
potentially exotic ) for all knots K; we show that these
satellite operators are distinct from all connected-sum operators, even up to
concordance, and that they induce bijective functions on topological
concordance classes of knots, as well as smooth concordance classes of knots
modulo the smooth 4--dimensional Poincare Conjecture.Comment: 20 pages, 9 figures; in the second version, we have added several new
results about surjectivity of satellite operators, and inverses of satellite
operators, and the exposition and structure of the paper have been improve
Pretzel links, mutation, and the slice-ribbon conjecture
Let p and q be distinct integers greater than one. We show that the
2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon
mutant, by using 3-fold branched covers and an obstruction based on Donaldson's
diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture
for 4-stranded 2-component pretzel links.Comment: 14 pages, 7 figures, V2: Implements suggestions from a referee
report. This version has been accepted for publication by MR
Concordance of knots in
We establish a number of results about smooth and topological concordance of
knots in . The winding number of a knot in is
defined to be its class in . We
show that there is a unique smooth concordance class of knots with winding
number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell
in the topological category. We say a knot in is slice (resp.
topologically slice) if it bounds a smooth (resp. locally flat) disk in
. We show that there are infinitely many topological concordance
classes of non-slice knots, and moreover, for any winding number other than
, there are infinitely many topological concordance classes even within
the collection of slice knots. Additionally we demonstrate the distinction
between the smooth and topological categories by constructing infinite families
of slice knots that are topologically but not smoothly concordant, as well as
non-slice knots that are topologically slice and topologically concordant, but
not smoothly concordant.Comment: 25 pages, 19 figures, final version, to appear in Journal of London
Mathematical Societ
On distinct finite covers of 3-manifolds
Every closed orientable surface S has the following property: any two
connected covers of S of the same degree are homeomorphic (as spaces). In this,
paper we give a complete classification of compact 3-manifolds with empty or
toroidal boundary which have the above property. We also discuss related
group-theoretic questions.Comment: 29 pages. V3: Implements suggestions from a referee report. This
version has been accepted for publication by IUM