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I am interested in low-dimensional topology and, in particular, interactions between 3-dimensional and 4-dimension manifolds, constructions of exotic 4-manifolds, and symplectic and contact geometry.
Recently, I have been exploring which negative definite plumbings of disk bundles over the 2-sphere with associated graphs containing exactly one cycle have boundaries that also bound rational homology circles. Such plumbings can be used to construct rational homology 3-spheres that bound rational homology 4-balls as well as exotic 4-manifolds via cut-and-paste.
In a related direction, I have also been exploring so-called χ-slice links, which are links in the 3-sphere that bound properly and smoothly embedded surfaces with no closed components and Euler characteristic 1 in the 4-ball. One motivation behind studying such links is that the double covers of the 3-sphere branched along χ-slice links with nonzero determinant are rational homology 3-spheres that bound rational homology 4-balls.
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| On χ-slice pretzel links.
Abstract
A link is called χ-slice if it bounds a smooth properly embedded surface in the 4-ball with no closed components and Euler characteristic 1.
If a link has a single component, then it is χ-slice if and only if it is slice.
This article aims to generalize known results about the sliceness of pretzel knots to the χ-sliceness of pretzel links. In particular, we completely classify positive and negative pretzel links that are χ-slice, and obtain partial classifications of 3-stranded and 4-stranded pretzel links that are χ-slice.
with Sophia Fanelle, Evan Huang, Ben Huenemann, Weizhe Shen, and Hannah Turner |
On slice quasi-alternating 3-braid closures.
Abstract
We complete the classification of all but two quasi-alternating 3-braid closures whose double branched covers bound rational homology balls. We then use this classification to prove a generalisation of the slice--ribbon conjecture for all non-alternating quasi-alternating 3-braid closures by explicitly constructing ribbon surfaces of Euler characteristic one bounded by such links. Our methods are based on the work of the second author on torus bundles that bound rational homology circles together with a Heegaard Floer refinement of the Donaldson's theorem sliceness obstruction due to Greene and Jabuka.
with Vitalijs Brejevs |
Geography of symplectic Lefschetz fibrations and rational blowdowns.
Abstract
We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic 4-manifolds. We present an example of a construction of a minimal symplectic exotic CP^2#-5CP^2 through this procedure applied to a genus-3 fibration.
with R. Inanc Baykur and Mustafa Korkmaz to appear in Transactions of the American Mathematical Society |
On the nonorientable 4-ball genus of torus knots.
Abstract
The nonorientable 4-ball genus of a knot K in the 3-sphere is the minimal genus of nonorientable surfaces in the 4-ball bounded by K. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable 4-ball genus of any knot. This bound is sharp for several families of torus knots, including T(4n,(2n±1)2) for all positive even integers n, a family Longo showed were counterexamples to Batson's conjecture. We also prove that, whenever p is an even positive integer and p/2 is not a perfect square, then the torus knot T(p,q) does not bound a locally flat Möbius band for almost all integers q relatively prime to p.
with Fraser Binns, Sungkyung Kang, and Paula Truöl |
Classification of torus bundles that bound rational homology circles.
Abstract
In this article, we completely classify torus bundles over the circle that bound 4-manifolds with the rational homology of the circle. Along the way, we classify certain integral surgeries along chain links that bound rational homology balls and explore a connection to 3-braid closures whose double branched covers bound rational homology 4-balls.
Algebraic & Geometric Topology 23-6 (2023), 2449--2518 |
Using rational homology circles to construct rational homology balls.
Abstract
Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology 3-spheres that bound rational homology 4-balls. In particular, we find infinite families of torus bundles over the circle that bound rational homology circles and provide a simple method for constructing more general plumbed 3-manifolds that bound rational homology circles. We then use these rational homology circles to show that, for example, -1-surgery along any twisted positively-clasped Whitehead double of any knot bounds a rational homology 4-ball and 1-surgery along any unknotting number one knot K with a positive crossing that can be switched to unknot K bounds a rational homology 4-ball.
Topology and its Applications, Vol. 291 (2021) |
Tight contact structures on some plumbed 3-manifolds.
Abstract
In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-manifolds due to Honda, we classify the tight contact structures on a certain class of plumbed 3-manifolds that bound non-simply connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of the Stein fillable contact structures.
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| Symplectically replacing plumbings with Euler characteristic 2 4-manifolds.
Abstract
We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define k-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic k 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-replaceable plumbing trees, and use one such tree to construct a symplectic exotic ℂP2#6(-ℂP2).
Journal of Symplectic Geometry, Vol. 18, No. 5 (2020), pp. 1285-1318 |
| Cut-and-paste operations and exotic 4-manifolds. |
| AMS Southeastern Sectional University of Southern Alabama |
October 2023 |
| Geometry Topology Working Seminar (2-talk series) Georgia Tech |
April 2023 |
| Knot Online Seminar Virtual |
April 2023 |
| Tech Topology Conference Georgia Tech |
December 2022 |
| Math Seminar Spelman College |
October 2022 |
| Math Club Agnes Scott College |
September 2022 |
| Topology Seminar University of Georgia |
April 2022 |
| AMS Southeastern Sectional Meeting University of Southern Alabama |
November 2021 |
| Research Horizons Seminar Georgia Tech |
November 2021 |
| Geometry Topology Working Seminar (3-talk series) Georgia Tech |
October 2021 |
| Algebra-Topology Seminar University of Alabama |
October 2021 |
| Geometry Topology Seminar Georgia Tech |
September 2021 |
| Low-dimensional topology and symplectic geometry weekend Remote |
April 2021 |
| Geometry Seminar University of Virginia |
December 2020 |
| Topology Seminar Brandeis University |
October 2020 |
| AMS Fall Western Sectional Meeting UC Riverside |
November 2019 |
| Geometry and Topology Seminar University of Massachusetts Amherst |
October 2019 |
| Moab Topology Conference Moab, UT |
May 2019 |
| Symplectic Geometry, Gauge Theory, and Categorification Seminar Columbia University |
April 2019 |
| Geometry and Topology Seminar University of Massachusetts Amherst |
September 2018 |
| Geometric structures on 3- and 4-manifolds Inter-University Centre, Dubrovnik, Croatia |
June 2018 |
| Joint Mathematics Meetings San Diego, CA |
January 2018 |
| Tech Topology Conference Georgia Tech |
December 2017 |
| Joint Mathematics Meetings Atlanta, GA |
January 2017 |
| Perspectives in Topology and Geometry of 4-manifolds Inter-University Centre, Dubrovnik, Croatia |
June 2016 |
| AMS Southeastern Sectional Meeting University of Georgia |
March 2016 |
| Geometry Seminar University of Virginia |
November 2015 |
| AMS Southeastern Sectional, University of Southern Alabama | October 2023 |
| Tech Topology Summer School, Georgia Tech | July 2023 |
| Low-dimensional Topology Workshop, Renyi Institute | March 2023 |
| Tech Topology Conference, Georgia Tech | December 2022 |
| AMS Southeastern Section Meetings, Chattanooga, TN | October 2022 |
| Tech Topology Conference, Georgia Tech | December 2021 |
| AMS Western Section Meetings, UC Riverside, CA | November 2019 |
| Knot concordance and low-dimensional manifolds, Le Croisic, France | June 2019 |
| Moab Topology Conference, USU Moab | May 2019 |
| Virginia Topology Conference, University of Virginia | December 2018 | Knotted Surfaces in 4-manifolds, UMass Amherst | October 2018 | The topology and geometry of low-dimensional manifolds, UT-Austin | July 2018 | Geometric Structures on 3- and 4-manifolds, Dubrovnik, Croatia | June 2018 | Joint Mathematics Meetings, San Diego, CA | January 2018 |
| Low-Dimensional Topology Conference, UCLA | January 2018 |
| Tech Topology Conference, Georgia Tech | December 2017 |
| 58th Annual Texas Geometry and Topology Conference, UT - Austin | November 2017 |
| 2017 Virginia Topology Conference, University of Virginia | November 2017 |
| CMO-BIRS: Low Dimensional Topology and Gauge Theory, Oaxaca, MX | August 2017 |
| Kylerec Workshop, Truckee, CA | May 2017 |
| Joint Mathematics Meetings, Atlanta, GA | January 2017 |
| Virginia Topology Conference, University of Virginia | November 2016 |
| Low Dimensional Topology Workshop, Central European University, Budapest, Hungary | July 2016 |
| Perspectives in Topology and Geometry of 4-manifolds, Dubrovnik, Croatia | June 2016 |
| Knots in the Triangle (Knots in Washington XLII), North Carolina State University | April 2016 |
| Graduate Student Topology & Geometry Conference, Indiana University | April 2016 |
| AMS Southeastern Sectional Meeting, University of Georgia | March 2016 |
| Joint Mathematics Meetings, Seattle, WA | January 2016 |
| William Rowan Hamilton Geometry and Topology Workshop, Trinity College, Dublin, Ireland | August 2015 |
| Princeton Low Dimensional Topology Workshop, Princeton University | June 2015 |
| Redbud Topology Conference, Oklahoma State University | April 2015 |
| Topology Student Workshop, Georgia Tech | June 2014 |
| Graduate Student Topology & Geometry Conference, UT - Austin | April 2014 |
| Tech Topology Conference, Georgia Tech | December 2013 |