Seminars and Colloquia by Series

Moebius bands in S^1xB^3 and the square peg problem by Peter Feller

Series
Geometry Topology Seminar
Time
Wednesday, April 3, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Feller – ETH Zurich –

Following an idea of Hugelmeyer, we give a knot theory reproof of a theorem of Schnirelman: Every smooth Jordan curve in the Euclidian plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.

Our main knot theory result is that the torus knot T(2n,1) in S^1xS^2 does not arise as the boundary of a locally-flat Moebius band in S^1xB^3 for square-free integers n>1. For context, we note that for n>2 and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the T(9,10) torus knot in S^3.

Based on joint work with Marco Golla.

Embedding Seifert fibered spaces in the 4-sphere

Series
Geometry Topology Seminar
Time
Monday, April 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ahmad Issa – University of Texas, Austin

Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.

Joint GT-UGA Seminar at UGA - A spectral sequence from Khovanov homology to knot Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 25, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Nathan Dowlin – Dartmouth
Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Using 2-torsion to obstruct topological isotopy

Series
Geometry Topology Seminar
Time
Monday, March 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Hannah Schwartz – Bryn Mawr
It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.

The nu+ equivalence class of genus one knots

Series
Geometry Topology Seminar
Time
Monday, March 4, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Kouki Sato – University of Tokyo
The nu+ equivalence is an equivalence relation on the knot concordance group. It is known that the equivalence can be seen as a certain stable equivalence on knot Floer complexes, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this talk, we show that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.

Joint GT-UGA Seminar at GT - Knots in homology spheres, concordance, and crossing changes

Series
Geometry Topology Seminar
Time
Monday, February 25, 2019 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris Davis – U Wisconsin Eau Claire
Any knot in $S^3$ may be reduced to a slice knot by making some crossing changes. Indeed, this slice knot can be taken to be the unknot. We show that the same is true of knots in homology spheres, at least topologically. Something more complicated is true smoothly, as not every homology sphere bounds a smooth simply connected homology ball. We prove that a knot in a homology sphere is null-homotopic in a homology ball if and only if that knot can be reduced to the unknot by a sequence of concordances and crossing changes. We will show that there exist knot in a homology sphere which cannot be reduced to the unknot by any such sequence. As a consequence, there are knots in homology spheres which are not concordant to those examples produced by Levine in 2016 and Hom-Lidman-Levine in 2018.

Joint GT-UGA Seminar at GT - Knot Traces and the Slice Genus

Series
Geometry Topology Seminar
Time
Monday, February 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lisa Piccirillo – UT Austin
Smooth simply connected 4-manifolds can admit second homology classes not representable by smoothly embedded spheres; knot traces are the prototypical example of 4-manifolds with such classes. I will show that there are knot traces where the minimal genus smooth surface generating second homology is not of the canonical type, resolving question 1.41 on the Kirby problem list. I will also use the same tools to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Heegaard Floer and the homology cobordism group

Series
Geometry Topology Seminar
Time
Monday, February 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jen Hom – Georgia Tech
We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

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