This is the website for the joint topology seminar between Duke, NCSU and UNC. The seminar will normally meet on selected Tuesdays with the location rotating between Duke, NCSU and UNC. An informal pretalk seminar aimed at graduate students will also be held before some talks.

Previous semesters: Spring 2017, Fall 2016.

 Speaker Date/Time Location Title Paul Melvin(Bryn Mawr College) Sept 1 3:30pm Duke Physics 205 "One is enough" Slava Krushkal(University of Virginia) Oct 2 4:30pm NCSU SAS 1102 Flow and Yamada polynomials, planar triangulations, and TQFT Peter Samuelson(University of Edinburgh) Oct 10 4:30pm UNC Phillips 383 Hall algebras and the Fukaya category Liam Watson(University of Glasgow) Oct 17 4:30pm NCSU SAS 2102 Modules from Heegaard Floer theory as curves in a punctured torus Kyle Hayden(Boston College) Oct 23 3:15pm Duke Physics 119 Complex curves through a contact lens Faramarz Vafaee(California Institute of Technology) Nov 7 4:30pm NCSUSAS 2102 The prism manifold realization problem Eylem Yildiz(Michigan State University) Nov 28 4:30pm NCSUSAS 4201 A note on knot concordance

# Abstracts

#### September 1, 2017 at 3:30 pm

Paul Melvin

"One is enough"

Abstract: Using Gabai's recent proof of the 4-dimensional light-bulb theorem, we show that any two homologous surfaces of the same genus embedded with simply-connected complements in a smooth 4-manifold become smoothly isotopic after one stabilization (connected summing with a 2-sphere bundle over the 2-sphere). This is joint work with Auckly, Kim, Ruberman and Schwartz.

#### October 2, 2017 at 4:00 pm

Slava Krushkal

Flow and Yamada polynomials, planar triangulations, and TQFT

Abstract: In the 1960s Tutte observed that the value of the chromatic polynomial of planar triangulations at (golden ratio +1) obeys a number of remarkable properties. In this talk I will explain how TQFT gives rise to a conceptual framework for studying planar triangulations. I will discuss several extensions of Tutte's results and applications to the structure of the chromatic and flow polynomials of graphs, and the Yamada polynomial of spatial graphs. This talk is based on joint works with Ian Agol and with Paul Fendley.

#### October 10, 2017 at 4:30 pm

Peter Samuelson

Hall algebras and the Fukaya category

Abstract: The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)

#### October 17, 2017 at 4:30 pm

Liam Watson

Modules from Heegaard Floer theory as curves in a punctured torus

Abstract: Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic ones. I will discuss a geometric interpretation of this class of modules in terms of immersed curves in the punctured torus. This point of view has some surprising consequences for closed three-manifolds that follow from simple combinatorics of curves. For example, one can show that, if the dimension of an appropriate version of the Heegaard Floer homology (of a rational homology sphere) is less than 5, then the manifold does not contain an essential torus. Said another way, this gives a certificate that the manifold admits a geometric structure à la Thurston. This is joint work with Jonathan Hanselman and Jake Rasmussen.

#### October 23, 2017 at 3:15 pm

Kyle Hayden

Complex curves through a contact lens

Abstract: Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

#### November 7, 2017 at 4:00 pm

Faramarz Vafaee

The prism manifold realization problem

Abstract: The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in the three-sphere. In recent years, the realization problem for C, T, O, and I-type spherical manifolds has been solved, leaving the D-type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized by a pair of relatively prime integers p > 1 and q. We determine a complete list of prism manifolds P(p, q) that can be realized by positive integral surgeries on knots in the three-sphere. The methodology undertaken to obtain the classification relies on tools from Floer homology and lattice theory, and is primarily combinatorial in nature. This is joint work with Ballinger, Ni, and Ochse.

#### November 28, 2017 at 4:30 pm

Eylem Yildiz

A note on knot concordence

Abstract: I will discuss concordances of knots in 3-manifolds. In particular I will show that all the knots in the free homotopy class of $S^1\times pt$ in $S^1 \times S^2$ are concordant to each other. By Akbulut it turns out that many of these concordances are invertible.