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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 …

Does Bias Have Shape? An Examination Of The Feasibility Of Algorithmic Detection Of Unfair Bias Using Topological Data Analysis, Ansel Steven Tessier

Senior Projects Spring 2022

Artificial intelligence and machine learning systems are becoming ever more prevalent; at every turn these systems are asked to make decisions that have lasting impacts on peoples’ lives. It is becoming increasingly important that we ensure these systems are making fair and equitable decisions. For decades we have been aware of biased and unfair decision making in many sectors of society. In recent years a growing body of evidence suggests these biases are being captured in data that are then used to build artificial intelligence and machine learning systems, which themselves perpetuate these biases. The question is then, can we …

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

Publications and Research

No abstract provided.