Research

BOUNDS IN THE SIMPLE HEXAGONAL LATTICE AND CLASSIFICATION OF 11-STICK KNOTS

with  Yueheng Bao, Ari Benveniste, Nicholas Cazet, Ansel Goh, Jiantong Liu, Ethan Sherman

To appear,  Journal of Knot Theory and its Ramifications (2024) ArXiv

 The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot and the figure-eight knot

VERTEX DISTORTION DETECTS THE UNKNOT

with  Nicholas Cazet, David Crncevic, Todd Fellman, Phillip Kessler, Nikolas Rieke, Vatsal Srivastava, Luis Torres

Journal of Knot Theory and its Ramifications  (2022) ArXiv 

The first two authors introduced vertex distortion and showed that the vertex distortion of the unknot is trivial. It was conjectured that the vertex distortion of a knot is trivial if and only if the knot is trivial. We will use Denne-Sullivan's bound on Gromov distortion to bound the vertex distortion of nontrivial lattice knots. We will then conclude that trivial vertex distortion implies the unknot, proving the conjecture. Additionally, the first conjecture in vertex distortion's debut article is proven and a vertex distortion calculator is given.

THE GEOGRAPHY AND ELECTION OUTCOME (GEO) METRIC: AN INTRODUCTION 

with Tommy Ratliff, Stephanie Somersille and Ellen Veomett

Election Law Journal (2022) ArXiv 

We introduce the Geography and Election Outcome (GEO) metric, a new method for identifying potential partisan gerrymanders. In contrast with currently popular methods, the GEO metric uses both geographic information about a districting plan as well as election outcome data, rather than just one or the other.

VERTEX DISTORTION OF LATTICE KNOTS

with Nicholas Cazet 

Journal of Knot Theory and its Ramifications (2022) ArXiv 

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the l_1-norm. We show that the vertex distortion of a lattice knot is 1 only if it is the unknot, and that there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

THE DISK COMPLEX AND TOPOLOGICALLY MINIMAL SURFACES IN THE 3-SPHERE

with Luis Torres 

Journal of Knot Theory and its Ramifications (2021) ArXiv 

We show that  genus g>1 Heegaard surfaces for the 3-sphere are topologically minimal with index 2g−1.

KIRBY-THOMPSON DISTANCE FOR TRISECTIONS OF KNOTTED SURFACES 

with Ryan Blair, Scott A. Taylor and Maggy Tomova 

Journal of the London Mathematical Society (2022) ArXiv 

We define an integer invariant L(T ) of a bridge trisection T of a smooth surface in the 4-sphere.  We show that when the invariant is zero, then the surface is unknotted. 

CHANGE COMES FROM WITHOUT: LESSONS LEARNED FROM A CHAOTIC YEAR

with Wes Maciejewski , John Bragelman, Tim Hsu, Andrea Gottlieb, Jordan Schettler, Trisha Bergthold & Bem Cayco

Problems, Resources, and Issues in Mathematics Undergraduate Studies (2020) 

Our university is one campus of the larger, 23-campus California State University system. In 2017, the Chancellor of the system discontinued the developmental mathematics programs at all 23 campuses. This article gives an overview of our response, highlighting the change our department underwent to improve our pre-calculus stream and general education courses, making them more interactive, supportive, and student-centered.

DISTORTION AND THE BRIDGE DISTANCE OF KNOTS

with Ryan Blair, Scott A. Taylor and Maggy Tomova 

Journal of Topology (2020) ArXiv 

We produce a lower bound on distortion from the bridge number and bridge distance of a knot.  

DECLINATION AS A METRIC TO DETECT PARTISAN GERRYMANDERING

with Tommy Ratliff and Ellen Veomett

Election Law Journal (2019) ArXiv 

We show that  genus g>1 Heegaard surfaces for the 3-sphere are topologically minimal with index 2g−1.

HYPERBOLIC MANIFOLDS CONTAINING HIGH TOPOLOGICAL INDEX SURFACES

with Matt Rathbun

Pacific Journal of Mathematics (2018) ArXiv 

We construct, for any natural number n, a hyperbolic manifold containing a surface of topological index n.

NEIGHBORS OF KNOTS IN THE GORDIAN GRAPH 

with Ryan Blair, Jesse Johnson, Scott A. Taylor and Maggy Tomova 

The American Mathematical Monthly (2017) ArXiv 

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance.  

EXCEPTIONAL AND COSMETIC SURGERIES ON KNOTS

with Ryan Blair, Jesse Johnson, Scott A. Taylor and Maggy Tomova 

Mathematische Annalen (2017) ArXiv

We show that the distance of a link K with respect to a bridge surface of any genus determines a lower bound on the genus of essential surfaces and Heegaard surfaces in the manifold that results from non-trivial Dehn surgery. 

DISTANCE TWO LINKS

with Ryan Blair, Jesse Johnson, Scott A. Taylor and Maggy Tomova

Geometriae Dedicata (2015) ArXiv

We characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. 

HIGH DISTANCE KNOTS IN CLOSED 3-MANIFOLDS

with Matt Rathbun

Journal of Knot Theory and its Ramifications (2012) ArXiv

We show that after a single stabilization, some core of a Heegaard splitting is arbitrarily high distance with respect to the splitting surface.