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: Fall 2016.

 Speaker Date/Time Location Title Peter Lambert-Cole(Indiana University) Feb 21 4:15pm NCSU SAS 1102 Conway mutation and knot Floer homology Michael Willis(University of Virginia) Mar 3 3:15pm Duke Physics 119 The Khovanov homology of infinite braids Laura Starkson(Stanford University) Mar 6 3:15pm Duke Physics 119 Manipulating singularities of Weinstein skeleta Dorothy Buck(Imperial College London) Apr 4 4:30pm Duke Physics 119 Knotted DNA Hoel Queffelec(CNRS and Institut Montpelliérain Alexander Grothendieck) Apr 13 4:30pm UNC Phillips 332 Around Chebyshev's polynomial and the skein algebra of the torus Alex Zupan(University of Nebraska-Lincoln) Apr 25 3:00pm NCSU SAS 1102 The Andrews-Curtis Conjecture and new handle decompositions of the 4-sphere

# Abstracts

#### Feburary 21, 2017 at 4:15 pm

Peter Lambert-Cole

Conway mutation and knot Floer homology

Abstract: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles.

#### March 3, 2017 at 3:15 pm

Michael Willis

The Khovanov homology of infinite braids

Abstract: In this talk, I will show that the limiting Khovanov chain complex of any inﬁnite positive braid categoriﬁes the Jones-Wenzl projector, extending Lev Rozansky's work with inﬁnite torus braids. I will also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general inﬁnite braids will also be considered. This is joint work with Gabriel Islambouli.

#### March 6, 2017 at 3:15 pm

Laura Starkson

Manipulating singularities of Weinstein skeleta

Abstract: Weinstein manifolds are an important class of symplectic manifolds with convex ends/boundary. These 2n dimensional manifolds come with a retraction onto a core n-dimensional stratified complex called the skeleton, which generally has singularities. The topology of the skeleton does not generally determine the smooth or symplectic structure of the 2n dimensional Weinstein manifold. However, if the singularities fall into a simple enough class (Nadler’s arboreal singularities), the whole Weinstein manifold can be recovered just from the data of the n-dimensional complex. We discuss work in progress showing that every Weinstein manifold can be homotoped to have a skeleton with only arboreal singularities (focusing in low-dimensions). Then we will discuss some of the expectations and hopes for what might be done with these ideas in the future.

#### April 4, 2017 at 4:30 pm

Dorothy Buck

Knotted DNA

Abstract: The central axis of the famous DNA double helix is often topologically constrained or even circular. The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function is to change the DNA axis topology -- for example converting a torus link into an unknot. Additionally, there are several protein families that change the axis topology as a by-product of their interaction with DNA. This talk will describe typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how Dehn surgery and other low-dimensional topological methods have been useful in understanding certain DNA-protein interactions, and discuss the most common topological techniques used to attack these problems.

#### April 13, 2017 at 4:30 pm

Hoel Queffelec

Around Chebyshev's polynomial and the skein algebra of the torus

Abstract: The diagrammatic version of the Jones polynomial, based on the Kauffman bracket skein module, extends to knots in any 3-manifold. In the case of thickened surfaces, it can be endowed with the structure of an algebra by stacking. The case of the torus is of particular interest, and C. Frohman and R. Gelca exhibited in 1998 a basis of the skein module for which the multiplication is governed by the particularly simple "product-to-sum" formula. I'll present a diagrammatic proof of this formula that highlights the role of the Chebyshev's polynomials, before turning to categorification perspectives and their interactions with representation theory. (This is joint work with H. Russell, D. Rose, and P. Wedrich)

#### April 25, 2017 at 3:00 pm

Alex Zupan

The Andrews-Curtis Conjecture and new handle decompositions of the 4-sphere

Abstract: The Andrews-Curtis Conjecture, proposed in the 1960s, asserts that every balanced presentation of the trivial group can be simplified with a set of moves, called Andrew-Curtis moves. Every handle decomposition of the 4-sphere with no 3-handles induces such a presentation, with handle-slides corresponding to Andrews-Curtis moves. The most prominent examples in this setting are due to Gompf-Scharlemann-Thompson, building off work of Akbulut-Kirby. We describe a new construction that generalizes the work of Gompf-Scharlemann-Thompson, with intriguing connections to the Andrews-Curtis Conjecture. This is joint work in progress with Jeffrey Meier.
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