Undergraduate research in mathematics is a collaborative apprenticeship where you investigate questions whose answers are not found in the back of a textbook. In my research group, our focus centers on low-dimensional topology—the study of knots, surfaces, 3-manifolds, and 4-manifolds. Because the full theoretical machinery of smooth 4-manifold topology requires substantial graduate-level algebraic and differential topology, undergraduate projects typically emphasize concrete structural models, combinatorial representations (such as knot diagrams, grid diagrams, and trisection diagrams), and computational experimentation.

Projects generally fall into two categories: theoretical inquiry and computational topology. Theoretical projects examine structural and combinatorial properties of knots, surfaces, or decompositions, requiring careful proof-writing, geometric visualization, and literature synthesis. Computational projects use Python or symbolic packages to generate examples, compute topological invariants, test conjectures on knot tables, or search for counterexamples. (For concrete project workflows and sample coding tasks, see the companion guide on Computational Tasks in Low-Dimensional Topology.) You do not need to be an expert in both areas; students with an interest in pure mathematics can pursue structural proofs, while students with programming backgrounds can focus on algorithm development and symbolic computation.

To begin research, you do not need advanced coursework in topology or differential geometry, but you do need mathematical curiosity and the perseverance to tackle problems that cannot be solved in a single sitting. The primary prerequisite is comfort with mathematical reasoning and proof techniques, typically gained through Discrete Mathematics, Foundations of Mathematics, or Linear Algebra. For computational projects, familiarity with Python, data structures, or symbolic computation packages is helpful. More important than specific course numbers is a willingness to read unfamiliar mathematics, learn software tools independently, and grapple with open-ended questions.

Undergraduate research requires a sustained weekly commitment comparable to a rigorous upper-level course—roughly 6 to 10 hours per week of reading, calculating, coding, and writing. We will meet weekly for one hour to discuss progress, troubleshoot obstacles, and set concrete goals for the following week. Our work often culminates in formal deliverables, such as presentations at the Winona State Ramaley Research Celebration or the MAA North Central Section meeting, an undergraduate thesis, or contributions to co-authored research manuscripts.

If you are interested in exploring a project, send me an email to schedule an initial conversation. In your email, include a brief summary of your mathematical and programming background, the courses you have completed or are currently taking, why you are interested in doing research in topology, and whether you are aiming for theoretical work, computational work, or a hybrid of both. We will arrange a time to discuss potential topics and find a project that aligns with your preparation and academic goals.

Past Student Research

Reviewing past student work offers a concrete look at how projects develop from foundational readings into specialized theses and presentations:

  • Parker Smiley — Vanishing Obstructions to Sliceness of Weaving Links (Master’s Thesis & Independent Study in Algebraic Topology). Parker’s Thesis

    Investigated alternating weaving links $W(p, q)$ and analyzed when classical obstructions to sliceness vanish. The project combined algebraic topology techniques with explicit Seifert surface constructions and combinatorial link invariants.

  • Bethany Harms — Knot and Link Polynomials (Master’s Project). Bethany’s Project

    Explored polynomial invariants of knots and links, focusing on skein relations for the the Alexander–Conway polynomial, the Jones polynomial, and the HOMFLY-PT polynomial.

Potential Research Projects

Project 1: Combinatorial Obstructions to Knot Sliceness

A knot $K \subseteq S^3$ is slice if it bounds a smoothly embedded disk in the 4-ball $B^4$. Determining whether a given knot is slice is a central problem in low-dimensional topology that bridges dimension three and dimension four. While modern gauge-theoretic invariants provide deep slice obstructions, several classical and semi-classical invariants—such as the Alexander polynomial via the Fox–Milnor condition, the Levine–Tristram signature function, and the Arf invariant—are combinatorial and computable directly from planar diagrams or Seifert surfaces. In this project, a student will investigate specific infinite families of knots (such as generalized twist knots, pretzel knots, or weaving links) to analyze when these classical obstructions vanish. Students will learn the topological foundations of 4-ball genus, construct Seifert matrices, compute signature jumps, and determine whether candidate slice knots admit explicit ribbon disk representations.

Key references for this project include:

  • Dale Rolfsen, Knots and Links (Publish or Perish, 1976; AMS Chelsea Publishing). Covers Seifert surfaces, Seifert matrices, the Alexander polynomial, and signature invariants.
  • Charles Livingston, Knot Theory (Carus Mathematical Monographs, Mathematical Association of America, 1993). An accessible treatment of slice knots, 4-ball genus, the Fox–Milnor condition, and algebraic concordance.
  • W. B. Raymond Lickorish, An Introduction to Knot Theory (Graduate Texts in Mathematics, Springer, 1997). Details Seifert pairings, quadratic forms, and classical signatures.

Project 2: Diagrammatics of Knotted Surfaces in the Four-Sphere

Visualizing 2-dimensional surfaces embedded in 4-dimensional space presents significant geometric challenges. Bridge trisections and tri-plane diagrams provide a combinatorial framework for studying knotted surfaces in $S^4$ by decomposing them into trivial 1-dimensional arcs across three coordinate half-spaces, governed by three interconnected planar link diagrams. This project explores diagrammatic complexity measures—such as minimal bridge index, crossing number, and tri-plane Reidemeister moves—for families of standard surfaces, including spun knots and non-orientable unknotted surfaces. A student working on this topic will develop facility with 4-dimensional diagrammatic calculus, track isotopic deformations across projection planes, and seek minimal crossing or bridge bounds for low-complexity knotted surfaces.

Key references for this project include:

  • John Hempel, 3-Manifolds (Annals of Mathematics Studies, Princeton University Press, 1976; AMS Chelsea Publishing). Establishes foundational handlebody decompositions and Heegaard splittings that motivate trisections and higher-dimensional diagrams.
  • J. Scott Carter and Masahico Saito, Knotted Surfaces and Their Diagrams (Mathematical Surveys and Monographs, American Mathematical Society, 1998). The classical treatise on visualizing and encoding surfaces in $\mathbb{R}^4$ via broken surface diagrams and motion pictures.
  • Jeffrey Meier and Alexander Zupan, Bridge trisections of knotted surfaces in $S^4$, I: Constrictions and bridge position, Journal of Topology 10 (2017), no. 3, 720–764. The foundational paper establishing tri-plane diagrams and bridge trisections.

Project 3: Classical Concordance and Band Pass Moves

A central question in knot concordance is understanding which geometric and diagrammatic invariants obstruct a knot from bounding an embedded disk in the 4-ball $B^4$. The band pass move (or pass move) is a classical local diagrammatic operation that passes a pair of oppositely oriented strands through another pair of oppositely oriented strands. By a theorem of Kauffman, pass equivalence partitions all classical knots into exactly two equivalence classes determined completely by the Arf invariant: knots pass equivalent to the unknot ($\operatorname{Arf}(K) = 0$) and those pass equivalent to the trefoil ($\operatorname{Arf}(K) = 1$). Because the Arf invariant is a classical concordance invariant, any two concordant knots are necessarily band pass equivalent. In this project, a student will examine the interplay between band pass moves, mod 2 Seifert forms, and classical concordance invariants. The work entails constructing explicit pass move sequences on knot diagrams, computing the Arf invariant via Seifert matrices and the Alexander polynomial, and analyzing how band pass equivalence interacts with algebraic concordance across designated families of knots.

Key references for this project include:

  • Louis H. Kauffman, Formal Knot Theory (Princeton University Press, 1983; Dover Publications). Introduces diagrammatic state models, formal pass equivalence, and the combinatorial development of the Arf invariant.
  • Louis H. Kauffman, On Knots (Annals of Mathematics Studies, Princeton University Press, 1987). Comprehensive exposition of the Arf invariant, link equivalence relations, and band operations.
  • Dale Rolfsen, Knots and Links (AMS Chelsea Publishing). Chapters 8 and 10 provide the topological framework for knot concordance, slice knots, and mod 2 intersection forms.