Georgia Institute of Technology
December 6 to 8, 2013


  Talk: Robert Lipshitz*   The Jones polynomial as Euler characteristic
Scanned Notes
  Social Gathering   Cypress Street Pint & Plate
  Light refreshments    
  Talk: Tim Cochran   Counterexamples to Kauffman's conjectures on slice knots
Scanned Notes
  Tea Break    
  Talk: Susan Williams   Links in thickened surfaces and virtual links
Slides from talk
  Talk: John Baldwin   From invariants of tangles to invariants of bordered 3-manifolds
Scanned Notes
  Extended Tea Break    
  Talk: Thomas Mark   Floer homology and fractional Dehn twists
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  Banquet   At: 4th and Swift
  Light refreshments    
  Talk: Alan Diaz   Braid isotopy, Khovanov homology and invariants of transverse knots
  Tea Break    
  Talk: Robert Gompf   Exotic smoothings of open 4-manifolds
Scanned Notes

All talks will be in Skiles 006.

* This will be a colloquium talk.

John Baldwin
Title: From invariants of tangles to invariants of bordered 3-manifolds
Abstract: I'll discuss work-in-progress with Jon Bloom toward developing an invariant of bordered 3-manifolds using monopole Floer homology. I'll focus on motivations for our construction stemming from Khovanov's work on a "functor-valued invariant of tangles". Hint: the relationship between tangles and bordered manifolds comes from taking branched double covers. Clarkson's recent results on the behavior of Floer homology under mutation means that our construction gives a faithful linear-categorical representation of the mapping class group of any closed surface of genus at least 3. The problem in genus 2 has to do with the hyperelliptic involution. I'll describe a strategy for using our bordered theory to show that mutation by this involution preserves the rank of Floer homology. If successful, our strategy would also imply that the rank of knot Floer homology is invariant under Conway mutation.

Tim Cochran
Title: Counterexamples to Kauffman's conjectures on slice knots
Abstract: In 1969 Jerome Levine introduced the philosophy that one could understand whether a knot is a slice knot by studying the closed curves on any Seifert surface. This allowed him to classify higher-odd-dimensional knot concordance. In 1982, in support of this philosophy, Louis Kauffman conjectured that if a knot is a slice knot then on any Seifert surface for that knot there exists an essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explicitly expressed in terms of invariants of curves on the Seifert surface. We give counterexamples to Kaufman's conjecture, that is, we exhibit (smoothly) slice knots that admit Seifert surfaces that contain no essential simple closed curve of self-linking zero with zero Arf invariant. Joint with Chris W. Davis.

Alan Diaz
Title: Braid isotopy, Khovanov homology and invariants of transverse knots
Abstract: In 2004, Plamenevskaya defined an invariant of transverse knots that takes the form of a distinguished class in the Khovanov homology of the smooth type of the knot. Work of Roberts, and Baldwin and Plamenevskaya, linked this invariant to the Heegaard Floer contact class of the induced contact structure on the branched double cover of the knot, via Ozsvath and Szabo's spectral sequence. This yielded combinatorial proofs of tightness or non-fillability for some of these contact manifolds. In another direction, several authors defined variations of Plamenevskaya's invariant in various generalizations of Khovanov homology. However, it was not known whether any of these invariants are strictly stronger than the classical self-linking number. I will show that most of these invariants, as well as the contact class of the branched double cover, are determined by self-linking. The key tool is a 2012 theorem of Dynnikov and Prasolov which implies that braid isotopies can be factored.

Robert Gompf
Title: Exotic smoothings of open 4-manifolds
Abstract: Smoothing theory for open 4-manifolds seems to have stagnated in the past decade or two, perhaps due to the misperception that since everything probably has uncountably many smoothings, there is nothing more to say. However, most traditional approaches involve tinkering with the end of the manifold without probing the deeper structure such as minimal genera of homology classes. We show that this genus function, together with its counterpart at infinity, can be controlled surprisingly well compared to the case of closed 4-manifolds, and these tools are often complementary to traditional techniques. We obtain many new smooth structures, including some on manifolds not previously known to admit exotic smoothings. This talk will include some material not covered in talks last summer.

Robert Lipshitz
Title: The Jones polynomial as Euler characteristic
Abstract: We will start by defining the Jones polynomial of a knot and talking about some of its classical applications to knot theory. We will then define a fancier version ("categorification") of the Jones polynomial, called Khovanov homology and mention some of its applications. We will conclude by talking about a further refinement, a Khovanov homotopy type, sketch some of the ideas behind its construction, and mention some applications. (This last part is joint work with Sucharit Sarkar.) At least the first half of the talk should be accessible to non-topologists.

Thomas Mark
Title: Floer homology and fractional Dehn twists
Abstract: Given a surface with connected boundary and a homeomorphism of the surface that is the identity on the boundary, one can define a rational number that measures the amount of "twisting around the boundary" induced by that homeomorphism. This number---the fractional Dehn twist coefficient---is invariant under isotopy relative to the boundary and under conjugation, and increases by 1 under composition with a boundary-parallel Dehn twist. Homeomorphisms fixing the boundary arise naturally as monodromies of fibered knots in 3-manifolds, hence we can associate a twist coefficient to any such fibered knot. The twist coefficient is known to have interesting connections with contact topology and with taut foliations on 3-manifolds. I will describe some joint work with Matthew Hedden that provides upper bounds for the absolute value of the twist coefficient of a fibered knot in terms of the Heegaard Floer homology of the ambient manifold, in particular proving that for any closed 3-manifold, the twist coefficient of any fibered knot in that manifold is bounded.

Susan Williams
Title: Links in thickened surfaces and virtual links
Abstract: We define new invariants of links in thickened surfaces $S \times I$. The covering group has a finite presentation as an operator group that is easily computed from a diagram; it conveniently encodes the fundamental group of the link exterior together with part of the peripheral structure. From it, we derive an Alexander-type polynomial invariant.

Virtual links may be regarded as links in thickened surfaces modulo isotopy and stabilization. The virtual genus is the minimal genus of a thickened surface supporting the link. We show that the covering group of any representative link determines the virtual genus, and obtain readily computable upper and lower bounds on virtual genus. (Joint work with J. Scott Carter and Daniel Silver)


organizers: J. Etnyre and D. Margalit
Supported by the NSF and the Georgia Institute of Technology