I am an Assistant Professor of Mathematics at the University of Central Missouri. I did my PhD at the University of Nebraska — Lincoln and was advised by Alex Zupan. My current interests are primarily in the geometric and algebraic topology of 3- and 4-manifolds. In particular, I study knot theory in dimensions 3 and 4, Heegaard splittings of 3-manifolds, and trisections of 4-manifolds. In addition, I am interested in the computational aspects of these areas, particularly computational algebraic topology and computational knot theory. A copy of my CV is available here.
2018-2024 PhD in Mathematics | ||
2018-2020 MS in Mathematics | ||
2015-2018 BS in Mathematics, Cum Laude; Minor: PhysicsExtracurricular Activities:
|
Suppose that $W$ and $W’$ are smooth, compact, and oriented $4$-manifolds that are either diffeomorphic to $S^1$ times the exterior $E_Y(K)$ of a fibered knot $K$ in a closed, connected, orientable $3$-manifold $Y$, or are diffeomorphic to $\Sigma_{g,1}$ bundles over the $2$-torus with monodromy fixing the boundary of the fiber pointwise. If $f: \partial W’ \to \partial W$ is an orientation-preserving diffeomorphism of the $3$-torus boundaries, we have that $X = W \cup_f W’$ is a closed, oriented $4$-manifold that fibers over $S^1$. In particular, if $W’ = T^2 \times D^2$ and $W = S^1 \times E_Y(K)$, then our result shows that the result of doing torus surgery in $S^1\times Y$ along $S^1 \times K$ is a $4$-manifold that fibers over $S^1$. Furthermore, we extend work of Zentner by showing that the result of torus surgery along $S^1$ times the unknot $\mathcal{U}$ in $S^1 \times S^3$ is diffeomorphic to $S^1$ times a lens space.
A compact $n$-manifold $X$ is fibered if it is a fiber bundle where the fiber $F$ and base space $B$ are manifolds. Fibered manifolds are particularly nice, as they are essentially classified by their monodromy maps. Two common examples of 4-dimensional fibered manifolds are surface bundles over surfaces and 3-manifold bundles over the circle.
The main focus of this dissertation is to investigate fibered 4-manifolds whose boundaries are the 3-torus and how these manifolds glue together to give new closed, fibered 4-manifolds. In particular, suppose W is diffeomorphic to $S^1\times E_Y(K)$ where $Y$ is a closed, oriented 3-manifold and $K$ is a fibered knot in $Y$, or that $W$ is diffeomorphic to a $\Sigma_{g,1}$-bundle over the torus, and let $W′$ be defined similarly. If $f:\partial W’ \to \partial W$ is an orientation-preserving diffeomorphism of the $T^3$ -boundary, we have that $X = W \cup_f W’$ fibers over the circle. We also study spun 4-manifolds and construct 4-secting Morse 2-functions on these manifolds. Suppose that $Y$ is a compact, oriented, connected 3-manifold with connected boundary $F = \partial Y$ and that $f:F \times S^1 \to F \times S^1$ is an orientation-preserving diffeomorphism. Then, we show that the $f$-spin of $Y$ admits a $(2g - h; g)$ 4-section if $h \neq -1$ or if $h = 1$ and $f$ is isotopic to the identity, where $h$ is the genus of $F$ and $g$ is the Heegaard genus of $Y$. This generalizes the work of Meier on trisections of spun 4-manifolds and of Kegel and Schmäschke on trisections of $4$-dimensional open book decompositions.
We define a pants distance for knotted surfaces in 4-manifolds which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the (4-manifold, surface) pair has a simple topology. Furthermore, we calculate the exact values of our invariants for many new examples such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.
Meier and Zupan proved that an orientable surface $\mathcal{K}$ in $S^4$ admits a tri-plane diagram with zero crossings if and only if $\mathcal{K}$ is unknotted. We determine the minimal crossing numbers of nonorientable unknotted surfaces in $S^4$, proving that $c(\mathcal{P}^{n,m}) = \max\{1,|n-m|\}$, where $\mathcal{P}^{n,m}$ denotes the connected sum of $n$ unknotted projective planes with normal Euler number $+2$ and $m$ unknotted projective planes with normal Euler number $-2$. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers.
Geometric topology is a subfield of topology that studies manifolds and maps between manifolds. Originating in roughly 1935, the past eighty-nine years have seen unprecedented growth and development, especially in low-dimensional topology, which studies nice shapes, called manifolds, in dimensions one through four.
In this talk, I will give an overview of one tool used to study topology in low dimensions, Morse theory. Particular focus will be given to Morse theory in dimensions one and two, which is essentially the study of critical points of functions of the form $y=f(x)$ and $z=f(x,y)$.
No background higher than Calculus I will be assumed, but this talk will be most accessible to those who have seen some Multivariable Calculus (Calculus III).
The Hopf surface is diffeomorphic to $S^1\times S^3$, and fibers over $S^2$ with fiber $T^2$. The result of doing torus surgery along an embedded $2$-torus $T$ of self-intersection zero in a closed $4$-manifold $X$ is the $4$-manifold $X_\varphi(T)$ obtained by removing a regular neighborhood of $T$ and replacing it with a copy of $T^2\times D^2$ along some diffeomorphism $\varphi$ of the resulting $2$-torus boundary.
Zentner showed that if the result of doing torus surgery along two fibers of the Hopf surface results in a homology Hopf surface, then the result is diffeomorphic to the standard Hopf surface. We extend these results to show that every manifold obtained by performing torus surgery along two fibers of the Hopf surface is diffeomorphic to $Y \times S^1$ where $Y$ is a $3$-manifold obtained by Dehn surgery along the unknot in $S^3$.
Last week, Kathryn started us on our exploration of group deficiency from a topological perspective. Today, I will pick up the torch and show how groups, 4-manifolds, and group deficiencies interact. In particular, I’ll state a theorem of A. A. Markov’s from the mid 1950s that gives a structural theorem on the fundamental groups of compact 4-manifolds. Then, I’ll dive into an exploration of the consequences of this theorem, namely how one can use homology of a compact 4-manifold to bound the deficiency of its fundamental group. Finally, if time permits, I’ll talk about recent work connecting torus surgery to group deficiency.
In this talk, I construct the Casson-Gordon signature invariants of a knot K. In doing so, we will take a voyage through an ocean of abstract nonsense, touching on twisted homology, cobordism groups, representation theory, and more!
Time permitting, we will also see how a particularly interesting subset of these 3-dimensional invariants were triviallized by the groundbreaking work in 4-dimensions of Micheal Freedman.
In this talk, we will go over the construction of Meier-Zupan generalized square knots and their associated R-links. We then will discuss how to turn an R-link into a balanced presentation of the trivial group, which yields many potential counterexamples to the Andrews-Curtis conjecture.
Last week, Audrey introduced us to the word problem (WP) for groups, and showed that residually finite (RF) groups have solvable WP. In this talk, I will outline how Thurston’s Geometrization Program (famously proved by Perelman) can be used to show that fundamental groups of compact 3-manifolds are residually finite, and hence have solvable WP. Along the way, we’ll acquire a “Who’s Who” gallery of important theorems in the theory of 3-manifolds and their fundamental groups.
In this talk, I will discuss Heegaard splittings of 3-manifolds and trisections of 4-manifolds. These decompositions cut the manifold into 1-handlebodies of appropriate dimension whose common intersection is a closed, orientable surface. By studying these decompositions, we obtain lots of information about the manifolds that they determine. This talk is based in part on David Gay’s “From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4.”
