Applying mathematics to the art of paper folding gives us the tools to
explore how a set of folds leads to a particular model. In this paper, we apply graph theory and geometry to origami to design an algorithm that can ...
In knot theory, a knot may have an invariant which is easily computed but difficult to understand geometrically and another invariant which is easily understood but quite difficult to compute. Knot theorists attempt to ...
In this paper, we present Artin’s original solution to the word
problem. We begin with a quick refresher on group presentations
via generators and relations in Section 2, and show how
nontrivial relations lead to the ...
The negative of the highest degree of the framing variable a in the Kauffman polynomial provides a bound for the Thurston-Bennequin number of Legendrian knots in a smooth knot type. This is a less than obvious connection ...
Knots and links—or embedded loops—can be understood by a variety of means, including diagrams in the plane and surfaces which they bound. By imposing constraints on the embedding of these loops, we obtain both geometric ...
The Riemann Mapping Theorem guarantees the existence of a conformal map between a simply connected region in C (that is not the entirety of C) and the unit disk, but it does not provide a means of constructing such a ...
This thesis is made up of two parts, which are connected by a common subject, Discrete Dynamical Systems. The first part treats the topic on a collegiate level, proving theorems about fixed points and their classification, ...
In this paper, I will introduce the mathematical ideas of knots and links, define a function on each that we call the energy of a knot or link, and prove several lemmas and theorems that relate the knot and link energies ...
An important consequence of a deep theorem by Kronheimer and Mrowka asserts that the 4-ball genus of any Legendrian knot is realized by a Lagrangian filling. Motivated by an open problem posed by Boileau and Fourrier, we ...
This senior thesis is made up of two parts, which are connected by
a common subject, hyperbolic transformational geometry. The first part, Hyperbolic Transformations in Two Models, treats the topic on a collegiate level, ...
This senior thesis is made up of two parts, which are connected by
a common subject, hyperbolic transformational geometry. The first
part, Hyperbolic Transformations in Two Models, treats the topic on a collegiate level, ...
This paper serves as an introduction to a field of topological graph theory, intrinsic knotting and linking of graphs. The motivation for this field is considering if graphs embedded into R³ contain subsets that are knots ...
An important type of research in knot theory attempts to relate two types of knot invariants, one which is easy to define but hard to compute and another which is easy to compute but difficult to understand geometrically. ...
The paper illustrates how concepts in Knot Theory are used to formulate
a mathematical model for the mechanism of the topoisomerase
TN3 Resolvase. The mathematical model allows for the proposed explanation
of the mechanism ...
This paper will explore several different definitions of fractal dimension.
Using some concrete examples, we will examine the relationships between
these definitions, and will prove that they do not always agree for ...
We describe results of Petters that use Morse theory to determine
properties of a generic single-plane gravitational lens system. In particular, we apply Morse theoretic tools to obtain lower bounds on the number of images ...
Homology gives a tool to measure the "holes" in topological spaces. Persistent homology extends the theory to define how homology changes over a filtration of a simplicial complex. Further it has been shown that persistent ...