Ken Baker
Bounding bridge numbers and hitting numbers
We highlight a technical lemma about intersecting surfaces in link exteriors and its use with annular twists of knots. This allows us to create, when a suitable auxiliary surface is present, a family of knots of arbitrarily large bridge number or hitting number. The former is due to work with Gordon-Luecke, the latter is due to ongoing work with Bowman-Luecke. We will discuss applications of both.
Ryan Blair
Knots with compressible thin levels
Recent advancements in the study of high-distance surfaces in knot exteriors have made it possible to construct knots in a way that puts heavy restrictions on all essential and strongly irreducible surfaces in the knot exterior. We use these techniques to construct the first examples of knots in Gabai thin position which admit compressible thin levels. This is joint work with Alex Zupan.
Sean Bowman
Bridge numbers of some twisted torus knots
We compute the bridge numbers of a family of famous twisted torus knots, those constructed by Morimoto-Sakuma-Yokota and shown by them to have genus one bridge number 2. The proof is part of a more general result on bridge numbers of twisted torus knots, joint with Scott Taylor and Alex Zupan.
Alexander Coward
Unknotting crossing changes and circular Heegaard Splittings
Abstract TBA
Mario Eudave-Muñoz
Tunnel number one knots and meridional surfaces
First, we remember a construction of (1,1)-knots in the 3-sphere that admit a meridional incompressible torus intersecting the knot in two points, and a more general construction of tunnel number one knots with the same property. Remember
that a (1,1)-knot has tunnel number one. In part, this is joint work with Enrique Ramirez-Losada. Then we will show that if K is a tunnel number one knot admitting a meridional incompressible torus intersecting it in two points, then K comes from one of the above constructions. This is joint work with Grissel Santiago-Gonzalez.
Cameron Gordon
Bridge number, Heegaard genus, and Dehn surgery
Let M be a 3-manifold obtained by some non-integral Dehn surgery
on a hyperbolic knot in S^3. Let K be the dual knot in M and let S be a
strongly irreducible Heegaard surface in M of genus g. We show that the
bridge number of K with respect to S is bounded above by a universal
linear function of g. As a consequence, there is a similar upper bound on
the Heegaard genus of the original knot exterior. In the case g=2 the
bounds that we obtain are optimal. This is joint work with Ken Baker and
John Luecke.
Jesse Johnson
Making progress in the curve complex
I will describe two relatively simple approaches to constructing geodesics in the curve complex, one loop at a time, in a
controlled manner. In other words, these techniques allow one to determine when a sequence of loops is consistently moving away from
its starting point. Moreover, I'll explain how these techniques fit in with the three-dimensional picture of Heegaard surface/bridge surface
distance.
Fabiola Manjarrez-Gutiérrez
Circular thin position, Morse-Novikov number and handle number.
Given a Morse map from a knot exterior to the circle we can recognize the critical points to obtain a decomposition of the knot exterior in such a way that the preimages of regular values are Seifert surfaces which are alternatively incompressible and weakly incompressible. This notion is known as circular thin position. There are other two invariants which rise from the study of circle-valued Morse maps, namely, the Morse-Novikov number which counts the minimum of critical points, and the handle number just counts the index-1 critical points. In this talk we will survey on recent results on circular thin position and its relations with Morse-Novikov number and handle number.
Makoto Ozawa
A knot with destabilized bridge spheres of arbitrarily high bridge number
We show that there exists an infinite family of knots each of which has, for each integer
k≥0, a destabilized
(2k+5)-bridge sphere. We also show that, for each integer
n≥ 4, there exists a knot with a destabilized
3-bridge sphere and a destabilized
n-bridge sphere. This is a joint work with Yeonhee Jang, Tsuyoshi Kobayashi and Kazuto Takao.
Jessica Purcell
Geometrically maximal knots
In this talk, we consider the ratio of volumes of hyperbolic knots to their crossing numbers. This ratio is known to have maximum value less than the volume of a regular ideal octahedron. This motivates several questions, such as, for which knots is the ratio very near the maximum? For fixed crossing number, what links maximize this ratio? We say that a sequence of hyperbolic knots is geometrically maximal if these ratios limit to the maximum value. In this talk, we describe several sequences of geometrically maximal knots, and present several conjectures. We discuss weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. We prove weaving knots are geometrically maximal. We discuss a method, using ideas known to Agol, for constructing other sequences of geometrically maximal knots. This is joint with Abhijit Champanerkar and Ilya Kofman.
Trent Schirmer
A lower bound on tunnel number degeneration
We use a combination of well known thin position techniques and a novel ``doppelganger'' construction to prove that t(K#K') is bounded below by max{t(K),t(K')}, where t(_) denotes tunnel number.
Scott Taylor
(A) width is additive
Thin position for knots in the 3-sphere provides a tangle decomposition of a knot which is similar in many respects to putting a knot into bridge position. Thin position gives rise to a knot invariant called "width", which is an analogue of the better known "bridge number". In light of Schubert's results concerning the (-1)-additivity of bridge number, it is natural to ask if width satisfies any additive properties under connected sum of knots. Blair and Tomova (2010) showed that width is not additive. It turns out, however, that if we change the definition of thin position and, correspondingly modify the definition of width, width does become additive. We survey a general theory which produces this result and which gives similar results for graphs in arbitrary closed, orientable 3-manifolds. This is preliminary work and is joint with Maggy Tomova.
Abby Thompson
Finding geodesics in a triangulated 2-sphere
Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory
to obtain a simple version of the three geodesics theorem for the 2-sphere. In general these three geodesics may be unstable,
corresponding, for example, to the three equators of an ellipsoid. Using a piece-wise linear approach, we show that we can usually
find at least three stable geodesics.
Anastasiia Tsvietkova
Intrinsic geometry and the invariant trace field of hyperbolic
3-manifolds
We will discuss a new connection between intrinsic geometry
of hyperbolic 3-manifolds and their commensurability invariants. In
particular, we will introduce complex parameters that capture geometric
information about intercusp geodesic arcs, and show that such parameters
generate the invariant trace field. We will show that for a tunnel
number k manifold it is enough to choose 3k specific parameters. For
many hyperbolic link complements, this approach allows one to compute
the field from a link diagram. This is a joint work with Walter Neumann,
based on earlier joint work with Morwen Thistlethwaite.
Alex Zupan
Bridge trisections of knotted surfaces
Recently, Gay and Kirby introduced a 4-dimensional analogue
of a Heegaard splitting called a trisection. We will adapt their
approach to show that every knotted surface in the 4-sphere admits a
bridge trisection; namely, there is a decomposition of the 4-sphere
into three 4-balls which splits the surface into a collection of
boundary parallel disks. This may be viewed as the 4-dimensional
version of a bridge splitting for a knot in the 3-sphere. We will
discuss lots of open questions and interesting directions for future
research. Much of this is joint work with Ken Baker, David Gay, and
Jeff Meier.
