Seminars and Colloquia by Series

Joint GT-UGA Seminar at GT: L-space surgeries and satellites by algebraic links

Series
Geometry Topology Seminar
Time
Monday, April 3, 2017 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sarah Rasmussen – University of Cambridge
Exploring when a closed oriented 3-manifold has vanishing reduced Heegaard Floer homology---hence is a so-called L-space---lends insight into the deeper question of how Heegaard Floer homology can be used to enumerate and classify interesting geometric structures. Two years ago, J. Rasmussen and I developed a tool to classify the L-space Dehn surgery slopes for knots in 3-manifolds, and I later built on these methods to classify all graph manifold L-spaces. After briefly discussing these tools, I will describe my more recent computation of the region of rational L-space surgeries on any torus-link satellite of an L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables. More generally, I will discuss the region of L-space surgeries on iterated torus-link satellites and algebraic link satellites, along with implications for conjectures involving co-oriented taut foliations and left-orderable fundamental groups.

Joint GT-UGA Seminar at GT: An integral lift of contact homology

Series
Geometry Topology Seminar
Time
Monday, April 3, 2017 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jo Nelson – Barnard College, Columbia University
I will discuss joint work with Hutchings which gives a rigorousconstruction of cylindrical contact homology via geometric methods. Thistalk will highlight our use of non-equivariant constructions, automatictransversality, and obstruction bundle gluing. Together these yield anonequivariant homological contact invariant which is expected to beisomorphic to SH^+ under suitable assumptions. By making use of familyFloer theory we obtain an S^1-equivariant theory defined with coefficientsin Z, which when tensored with Q recovers the classical cylindrical contacthomology, now with the guarantee of well-definedness and invariance. Thisintegral lift of contact homology also contains interesting torsioninformation.

(1,1) L-space knots

Series
Geometry Topology Seminar
Time
Monday, April 3, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Josh Greene – Boston College
I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.

Differential Algebra of Cubic Graphs

Series
Geometry Topology Seminar
Time
Monday, March 27, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roger Casals – MIT
In this talk we associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations. From there, we study the Legendrian surfaces behind these combinatorial constructions, including Legendrian surgeries and the count of Morse flow trees, and discuss the proof of the correspondence between augmentations and constructible sheaves for this class of Legendrians.

Joint GT-UGA Seminar at GT: Gluck twists and trisections

Series
Geometry Topology Seminar
Time
Monday, March 13, 2017 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Gay – UGA
This is joint work with Jeff Meier. The Gluck twist operation removes an S^2XB^2 neighborhood of a knotted S^2 in S^4 and glues it back with a twist, producing a homotopy S^4 (i.e. potential counterexamples to the smooth Poincare conjecture, although for many classes of 2-knots theresults are in fact known to be smooth S^4's). By representing knotted S^2's in S^4 as doubly pointed Heegaard triples and understanding relative trisection diagrams of S^2XB^2 carefully, I'll show how to produce trisection diagrams (a.k.a. Heegaard triples) for these homotopy S^4's.(And for those not up on trisections I'll review the foundations.) The resulting recipe is surprisingly simple, but the fun, as always, is in the process.

Involutive Heegaard Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 13, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen Hendricks – Michigan State University
We use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We also give a formula for how this theory behaves under connected sum, and use it to give examples not homology cobordant to anything computable via our surgery formula. This is joint work with C. Manolescu; the last part of is also joint with I. Zemke.

Cellular Legendrian contact homology for surfaces

Series
Geometry Topology Seminar
Time
Monday, March 6, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Rrutherford – Ball State University
This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our cellular DGA is equivalent to the Legendrian contact homology DGA of L whose construction was carried out in this setting by Etnyre-Ekholm-Sullivan with the differential defined by counting holomorphic disks in C2 with boundary on the Lagrangian projection of L. Equivalence of our DGA with LCH is established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.

Jones slopes and Murasugi sums of links

Series
Geometry Topology Seminar
Time
Monday, February 20, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christine Lee – University of Texas at Austin –
A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot. A link diagram D is said to be a Murasugi sum of two links D' and D'' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D'', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces. We consider near-adequate links which are certain Murasugi sums of near-alternating link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots.

0-concordance of 2-knots

Series
Geometry Topology Seminar
Time
Monday, February 13, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nathan Sunukjian – Calvin College
A 2-knot is defined to be an embedding of S^2 in S^4. Unlike the theory of concordance for knots in S^3, the theory of concordance of 2-knots is trivial. This talk will be framed around the related concept of 0-concordance of 2-knots. It has been conjectured that this is also a trivial theory, that every 2-knot is 0-concordant to every other 2-knot. We will show that this conjecture is false, and in fact there are infinitely many 0-concordance classes. We'll in particular point out how the concept of 0-concordance is related to understanding smooth structures on S^4. The proof will involve invariants coming from Heegaard-Floer homology, and we will furthermore see how these invariants can be used shed light on other properties of 2-knots such as amphichirality and invertibility.

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