Mathematics Commons^™

Higher Spanier Groups, Johnny Aceti

West Chester University Master’s Theses

When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …

Bicategorical Traces And Cotraces, Justin Barhite

Theses and Dissertations--Mathematics

Familiar constructions like the trace of a matrix and the Euler characteristic of a closed smooth manifold are generalized by a notion of trace of an endomorphism of a dualizable object in a bicategory equipped with a piece of additional structure called a shadow functor. Another example of this bicategorical trace, in the form of maps between Hochschild homology of bimodules, appears in a 1987 paper by Joseph Lipman, alongside a more mysterious ”cotrace” map involving Hochschild cohomology. Putting this cotrace on the same category-theoretic footing as the trace has led us to propose a ”bicategorical cotrace” in a closed …

Kōlams In Graph Theory: Mathematics In South Indian Ritual Art, Nathan Hartmann

Murray State Theses and Dissertations

Kōlams are a ritual art form found in India, most commonly in the southern state
of Tamil Nadu. Comprised of different interlocking knots, these women-drawn designs are placed on the entrances to people’s home to showcase the household’s emotional state and ask the earth goddess Bhūdevi for forgiveness. More aesthetically pleasing kōlams are considered latshanam, where the design permeates beauty; monolinearity is one such aspect that implements latshanam. Using graph theory, we examine one style of these drawings, the labyrinthine variety, to identify if a given kōlam is monolinear and how to construct monolinear kōlams.

Spectral Sequences And Khovanov Homology, Zachary J. Winkeler

Dartmouth College Ph.D Dissertations

In this thesis, we will focus on two main topics; the common thread between both will be the existence of spectral sequences relating Khovanov homology to other knot invariants. Our first topic is an invariant MKh(L) for links in thickened disks with multiple punctures. This invariant is different from but inspired by both the Asaeda-Pryzytycki-Sikora (APS) homology and its specialization to links in the solid torus. Our theory will be constructed from a Z^n-filtration on the Khovanov complex, and as a result we will get various spectral sequences relating MKh(L) to Kh(L), AKh(L), and APS(L). Our …

A Topologist’S Broken Heart, Josh Hiller

Journal of Humanistic Mathematics

A poem about a topologist's broken heart.

Mining The Soma Cube For Gems: Isomorphic Subgraphs Reveal Equivalence Classes, Edward Vogel, My Tram

Journal of Humanistic Mathematics

Soma cubes are an example of a dissection puzzle, where an object is broken down into pieces, which must then be reassembled to form either the original shape or some new design. In this paper, we present some interesting discoveries regarding the Soma Cube. Equivalence classes form aesthetically pleasing shapes in the solution set of the puzzle. These gems are identified by subgraph isomorphisms using SNAP!/Edgy, a simple block-based computer programming language. Our preliminary findings offer several opportunities for researchers from middle school to undergraduate to utilize graphs, group theory, topology, and computer science to discover connections between computation and …

Thickened Surfaces, Checkerboard Surfaces, And Quantum Link Invariants, Joseph W. Boninger

Dissertations, Theses, and Capstone Projects

This dissertation has two parts, each motivated by an open problem related to the Jones polynomial. The first part addresses the Volume Conjecture of Kashaev, Murakami, and Murakami. We define a polynomial invariant, J^T_n, of links in the thickened torus, which we call the nth toroidal colored Jones polynomial, and we show J^T_n satisfies many properties of the original colored Jones polynomial. Most significantly, J^T_n exhibits volume conjecture behavior. We prove a volume conjecture for the 2-by-2 square weave, and provide computational evidence for other links. We also give two equivalent constructions …

The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

Finite N-Quandles Of Twisted Double Handcuff And Complete Graph, Veronica Backer-Peral

Honors Thesis

The Double Handcuff and K4 graphs can be generalized to a single family of spatial graphs by adding a variable number of twists between two edges. We can identify spatial graphs by calculating a quotient of the fundamental quandle, known as an N-quandle, which is a spatial graph invariant. In this paper, we prove that the N-quandle associated with this family of spatial graphs is finite when all but two edges are given a label of 2, and the remaining two edges are assigned labels from the natural numbers. To prove that the N-quandle is finite, we produce Cayley graphs …

Algebraic Invariants Of Knot Diagrams On Surfaces, Ryan Martinez

HMC Senior Theses

In this thesis we first give an introduction to knots, knot diagrams, and algebraic structures defined on them accessible to anyone with knowledge of very basic abstract algebra and topology. Of particular interest in this thesis is the quandle which "colors" knot diagrams. Usually, quandles are only used to color knot diagrams in the plane or on a sphere, so this thesis extends quandles to knot diagrams on any surface and begins to classify the fundamental quandles of knot diagrams on the torus.

This thesis also breifly looks into Niebrzydowski Tribrackets which are a different algebraic structure which, in future …

Trapped Surfaces, Topology Of Black Holes, And The Positive Mass Theorem, Lan-Hsuan Huang, Dan A. Lee

Publications and Research

No abstract provided.

From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar

Mathematics Faculty Research Publications

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. …

Visualizing Geometric Structures On Topological Surfaces, Andrea Clark

All NMU Master's Theses

We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the …

Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M^n-1 in Euclidean space Eⁿ is proper Dupin if the number of distinct principal curvatures is constant on M^n-1, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together …

An Analysis And Comparison Of Knot Polynomials, Hannah Steinhauer

Senior Honors Projects, 2020-current

Knot polynomials are polynomial equations that are assigned to knot projections based on the mathematical properties of the knots. They are also invariants, or properties of knots that do not change under ambient isotopy. In other words, given an invariant α for a knot K, α is the same for any projection of K. We will define these knot polynomials and explain the processes by which one finds them for a given knot projection. We will also compare the relative usefulness of these polynomials.

A Structure Theorem For Bad 3-Orbifolds, Rachel Julie Lehman

Graduate Theses and Dissertations

We explicitly construct 10 families of bad 3-orbifolds, X , having the following property: given any bad 3-orbifold, O, it admits an embedded suborbifold X ∈ X such that after removing this member from O, and capping the resulting boundary, and then iterating this process finitely many times, you obtain a good 3-orbifold. Reversing this process gives us a procedure to obtain any possible bad 3-orbifold starting with a good 3-orbifold. Each member of X has 1 or 2 spherical boundary components and has underlying topological space S2 × I or (S2 × S1)\B3.

Clustering Methods For Gene Expression Data Of Oxytricha Trifallax, Kyle Houfek

USF Tampa Graduate Theses and Dissertations

Clustering is a data analysis method which is used in a large variety of research fields. Many different algorithms exist for clustering, and none of them can be considered universally better than the others. Different methods of clustering are expounded upon, including hierarchical clustering and k-means clustering. Topological data analysis is also described, showing how topology can be used to infer structural information about the data set. We discuss how one finds the validity of clusters, as well as an optimal clustering method, and conclude with how we used various clustering methods to analyze transcriptome data from the ciliate Oxytricha …

Complete Bipartite Graph Embeddings On Orientable Surfaces Using Cayley Maps, Madeline Spies

Honors Program Theses

We explored how effective Cayley Maps are at embedding complete bipartite graphs onto orientable surfaces, such as spheres and tori. We embedded the graphs onto surfaces using Cayley Maps with the intent of finding rotations that result in the graphs’ optimal genera. Because there are only three groups that can be used to describe Kp,p, where p is prime, we chose to focus on this specific graph type. This paper explains how we determined the genera when using a Cayley Map, provides general theorems for surface face sizes and the Dihedral Group, and discusses our results for Kp,p up to …

Periodic Points On Tori: Vanishing And Realizability, Shane Clark

Theses and Dissertations--Mathematics

Let $X$ be a finite simplicial complex and $f\colon X \to X$ be a continuous map. A point $x\in X$ is a fixed point if $f(x)=x$. Classically fixed point theory develops invariants and obstructions to the removal of fixed points through continuous deformation. The Lefschetz Fixed number is an algebraic invariant that obstructs the removal of fixed points through continuous deformation. \[L(f)=\sum_{i=0}^\infty (-1)^i \tr\left(f_i:H_i(X;\bQ)\to H_i(X;\bQ)\right). \] The Lefschetz Fixed Point theorem states if $L(f)\neq 0$, then $f$ (and therefore all $g\simeq f$) has a fixed point. In general, the converse is not true. However, Lefschetz Number is a complete invariant …

A Spider's Web Of Doughnuts, Daniel Stoertz

Graduate Research Theses & Dissertations

This dissertation studies an interplay between the dynamics of iterated quasiregular map-

pings and certain topological structures. In particular, the relationship between the Julia set

of a uniformly quasiregular mapping f : R 3 → R 3 and the fast escaping set of its associated

Poincaré linearizer is explored. It is shown that, if the former is a Cantor set, then the latter

is a spider’s web. A new class of uniformly quasiregular maps is constructed to which this

result applies. Toward this, a geometrically self-similar Cantor set of genus 2 is constructed.

It is also shown that for any …

Merge Trees In Discrete Morse Theory, Benjamin Johnson

Mathematics Summer Fellows

The field of topological data analysis seeks to use techniques in topology to study large data sets. The hope is that rather than single quantities that summarize the data, such as mean or standard deviation, information about the data can be learned by studying the overall ``shape” of the data. One way to summarize this data is through a merge tree. Merge trees can be thought of as keeping track of certain clusters of data and determining when they merge together. In this paper, we will study merge trees induced by a discrete Morse function on a tree. Under a …

Properties Of Functionally Alexandroff Topologies And Their Lattice, Jacob Scott Menix

Masters Theses & Specialist Projects

This thesis explores functionally Alexandroff topologies and the order theory asso- ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set.

The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies …

Connectedness- Its Evolution And Applications, Nicholas A. Scoville

Topology

No abstract provided.

An 1883 Faery Tale, Scott W. Williams

Journal of Humanistic Mathematics

A poem about the construction of Georg Cantor's famous set.

Cantor Sets, Cantorvals, And Their Topological Structure, Ángel Adrián Agüero

Open Access Theses & Dissertations

With interesting topological properties, the Cantor set is worth studying for itself. In other areas, topological structures arise that are in fact homeomorphic to the Cantor set. In particular, we see sets that are homeomorphic to the Cantor set which result from the subsums of particular series, as well as linear combinations of algebraic sums of Cantor sets. These also result in what has been termed a Cantorval, which we also investigate.

Selective Strong Screenability, Isaac Joseph Coombs

Boise State University Theses and Dissertations

Screenability and strong screenability were both introduced some sixty years ago by R.H. Bing in his paper Metrization of Topological Spaces. Since then, much work has been done in exploring selective screenability (the selective version of screenability). However, the corresponding selective version of strong screenability has been virtually ignored. In this paper we seek to remedy this oversight. It is found that a great deal of the proofs about selective screenability readily carry over to proofs for the analogous version for selective strong screenability. We give some examples of selective strongly screenable spaces with the primary example being Pol's …

Persistence Equivalence Of Discrete Morse Functions On Trees, Yuqing Liu

Mathematics Summer Fellows

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. We then compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. We conclude with an example illustrating our construction.

Nearness Without Distance, Nicholas A. Scoville

Topology

No abstract provided.

3-Maps And Their Generalizations, Kevin J. Mccall

Theses and Dissertations

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

An Introduction To Topology For The High School Student, Nathaniel Ferron

Masters Essays

No abstract provided.