Peter Feller (Boston College)
Title:Smooth slice genus vs.
topological slice genus.
Abstract:Smooth and topological
fourmanifolds are wellknown to be quite
different.A knot theoretic incarnation of
this is the difference between the smooth
and the topological slice genus. We
examine this difference from two points of
view. In the first part of the talk we
discuss general bounds for the slice
genera in terms of other knot invariants;
including the following result:the degree
of the Alexander polynomial is an upper
bound for the topological slice genus.In
the second part of the talk we focus on
two particularly simple classes of knots:
torus knots and twobridge knots.
Stefan Friedl (Regensberg)
Title: Fox calculus and the Thurston norm
Abstract: If N is a 3manifold such that
its fundamental group has a presentation
with two generators and one relator we
will show how to obtain the Thurston norm
of N from Fox calculus.
Jen Hom (Columbia)
Title: Surgery obstructions and Heegaard
Floer homology
Abstract: Lickorish and Wallace proved
that every closed oriented threemanifold
can be expressed as surgery on a link in
the threesphere. This naturally leads to
the question of which threemanifolds are
surgery on a knot. We give an obstruction
coming from Heegaard Floer homology, and
use it to construct infinitely many small
Seifert fibered integer homology spheres
which are not surgery on a knot. This is
joint work with Cagri Karakurt and Tye
Lidman.
Robion Kirby (Berkeley)
Title: Trisections of 4manifolds
Abstract: Topics that are
independent of David Gay's lectures in his
minicourse.
Slava Krushkal (Virginia)
Title: Engel relations in 4manifold
topology
Abstract: The 2Engel relation has been
classically studied in finite and Lie
groups, dating back to the work of
Burnside. I will describe new applications
in link theory and 4manifold topology,
focusing on the 4dimensional topological
surgery conjecture. (Joint work with
Michael Freedman)
Joan Licata (Australian NU)
Title: Morse Structures on Open Books
Abstract: Every contact 3manifold is
locally contactomorphic to the standard
contact R^3, but this fact does not
necessarily produce large charts that
cover the manifold efficiently. I'll
describe joint work with Dave Gay which
uses an open book decomposition of a
contact manifold to produce a particularly
efficient collection of such
contactomorphisms, together with simple
combinatorial data describing how to
reconstruct the contact 3manifold from
these charts. We use this
construction to define front
projections for Legendrian knots and links
in arbitrary contact 3manifolds,
generalising existing constructions of
front projections for Legendrian knots in
S^3 and universally tight lens
spaces.
Paolo Lisca (Pisa)
Title: On 3braid knots of finite
concordance order.
Abstract: The sliceribbon conjecture was
established for 2bridge knots and
infinite families of pretzel and
Montesinos knots. The methods leading to
these results were also used to determine
the smooth concordance order of 2bridge
knots and infinitely many pretzel knots.
After recalling basic definitions I will
describe the methods mentioned above, and
then sketch how to use them to prove that
a knot of finite smooth concordance order
which is the closure of a 3braid is
either ribbon or alternating.
Duncan McCoy (Glasgow)
Title: Alternating knots with unknotting
number one
Abstract: The unknotting number of a knot,
$K$, is a defined to be the minimal number
of crossing changes required in any
diagram of $K$ to obtain the unknot.
Despite its straightforward definition,
the unknotting number is frequently
difficult to compute. For alternating
knots it can be shown that a knot has
unknotting number one if and only if every
alternating diagram contains an unknotting
crossing. I will explain the proof of this
result, which uses an obstruction,
originally due to Greene, arising from
Donaldson's Theorem and Heegaard Floer
homology. This approach also shows that an
alternating knot has unknotting number one
if and only if its double branched cover
arises by halfinteger surgery on a knot
in $S^3$. I will also talk about how one
can extend this work to show that if the
branched double cover of an alternating
link arises as noninteger surgery on a
knot in $S^3$, then this is exhibited by a
rational tangle replacement in an
alternating diagram.
Frank Quinn (Virginia Tech)
Title: Some decompositions of smooth
4manifolds
Abstract: We describe several ways to
produce decompositions, including
simplified versions of the GayKirby
trisections, the author’s dual
decompositions, and descriptions as
singular 2parameter families of surfaces.
These are all extremely complicated and it
is not yet clear how they might be useful.
Better connections with analysis are
probaby needed for defining invariants.
Good questions that allow some changes in
the manifold might provide guidance.
Jennifer Schultens (Davis)
Title: Kakimizu complexes
Abstract: In the 1980's, Osamu Kakimizu
defined a complex associated with a
knot.The complex encodes isotopy classes
of Seifert surfaces. Several teams
of researchers studied the complex over
the intervening years. Most notably,
JohnsonWilsonPelayo showed that the
Kakimizu complex of a knot is
quasiEuclidean. The Kakimizu
complex can be defined more generally, for
3manifolds and also for surfaces. I
will discuss this more general
definition. I will show how the
Kakimizu complex of a surface compares to
and differs from the curve complex and
homology curve complex. I will
exhibit examples demonstrating that the
Kakimizu complex of a 3manifold is not,
in general, quasiEuclidean.
Saso Strle (Ljubljana)
Title: Essential nonorientable surfaces
in homology cobordisms and closed
4manifolds
Abstract: I will discuss constraints on
embeddings of a nonorientable surface in
a given homology class in a 4manifold.
These take the form of inequalities
involving the genus and normal Euler class
of the surface, and some HeegaardFloer
theoretic invariants. The methods apply to
embeddings in closed 4manifolds or in a
homology M × I, where M is a rational
homology 3sphere. In the latter case it
is interesting to compare the resulting
genus bounds with those in the 3manifold.
For example, it turns out that for lens
spaces the 3 and 4dimensional bounds
agree. This is joint work with Adam Levine
and Daniel Ruberman.
Peter Teichner (Max Planck & UC
Berkeley)
Title: Pulling apart 2spheres in 4space
Abstract: In joint work with Rob
Schneiderman, we completed the computation
of the group of link maps of two 2spheres
into 4space, up to link homotopy. The
proof uses standard techniques like the
Whitney move, symmetric surgery and
algebraic selfintersection invariants but
in addition introduces new 4manifold
tools like Whitney spheres and
constructions of accessory disks.
Abigail Thompson (UC Davis)
Title: Incompressible surfaces and
distance 2 manifolds
Abstract: Let M be a closed orientable
3manifold with minimal genus Heegaard
surface F. If the
distance of the Heegaard splitting is at
least 3, it is known that M is atoroidal
and hence hyperbolic. So
all toroidal manifolds have distance at
most 2; but not all distance 2 manifolds
are toroidal. We would like to
be able to sort out which are which, and
we give some results towards this
end. In particular, we show that if
such an M is toroidal, M either contains a
nicelypositioned incompressible surface
(with respect to a minimal genus Heegaard
surface), or M has a small Seifertfibered
space component in its JSJ decomposition,
or the minimal genus splitting of M is not
thin.
Alexander Zupan (University of
NebraskaLincoln)
Title: Trisections of 4manifolds and Dehn
surgery
Abstract: A trisection of a
4manifold X is a splitting of X into
three simple pieces that glue together in
a prescribed way. Every trisection
gives rise to a handle decomposition of
X. As such, it is often possible to
extract results about trisections from
known facts about Dehn surgery, and vice
versa. In this talk, we will explore
this relationship; in particular, we will
discuss connections between trisections
and different versions of the Generalized
Property R Conjecture. Much of this
talk is based on joint work with Jeffrey
Meier and Trent Schirmer.
