Dissertation
Constructing Legendrian invariants via Legendrian ribbon categories
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008328
Abstract
This work extends the connections between category theory and knot theory by constructing a categorical framework for constructing invariants for oriented, framed Legendrian knots and links, established in collaboration with B. Cooper. These results not only show that classical Legendrian knot invariants can be obtained from a categorical framework, but new invariants can also arise.
The first half of this thesis focuses on framed Legendrian tangles. First we define a way of diagrammatically depicting framed Legendrian tangles using planar diagrams which we call framed front diagrams. Then we provide a set of five diagrammatic moves that can be performed on these framed front diagrams which preserve the framed Legendrian isotopy classes. We call these diagrammatic moves the framed Legendrian Reidemeister moves. With these framed Legendrian Reidemeister moves, we are able to provide a presentation for the category of oriented, framed Legendrian tangles colored over a strict monoidal category $\C$. This presentation also results in presentations for three additional categories.
In the second half of this work, we define invariants for oriented, framed Legendrian knots and links by using categories we call Legendrian ribbon categories. We show how each object in a Legendrian ribbon category can be used to construct an invariant and provide many different examples of Legendrian ribbon categories. These Legendrian ribbon categories should be thought of as a generalization of ribbon categories found in \cite{RESH, turaev} because we prove every ribbon category is an example of a Legendrian ribbon category. The primary source of our examples of Legendrian ribbon categories come by manipulating the ribbon structure of the category of representations of ribbon Hopf algebras. With these manipulations we are able to utilizes more of the information in ribbon Hopf algebras to define invariants than has traditionally been used. Specifically, we show how the structure of the group algebra can be used to obtain invariants for oriented, framed Legendrian links that are not invariants for oriented, framed classical links.
Details
- Title: Subtitle
- Constructing Legendrian invariants via Legendrian ribbon categories
- Creators
- Matthew Lee Barber
- Contributors
- Benjamin Cooper (Advisor)Frauke Bleher (Committee Member)Mohammad Farajzadeh-Tehrani (Committee Member)Ryan Kinser (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008328
- Publisher
- University of Iowa
- Number of pages
- vi, 187 pages
- Copyright
- Copyright 2026 Matthew Lee Barber
- Language
- English
- Date submitted
- 04/28/2026
- Description illustrations
- illustrations, graphs
- Description bibliographic
- Includes bibliographical references (pages 185-187).
- Public Abstract (ETD)
A framed Legendrian knot may be visualized as a ribbon whose ends are glued together to form a closed loop, with the additional requirement that the centerline of the ribbon lies in three-dimensional space in a way that satisfies a geometric condition arising from contact topology. A framed Legendrian link is a collection of such ribbons that may be intertwined with one another. For a framed Legendrian link to be oriented, each ribbon must be assigned a direction of travel along its length.
Two oriented, framed Legendrian links are considered equivalent or the same if one can be continuously deformed into the other in three-dimensional space without cutting the ribbons, allowing them to pass through themselves, or violating the geometric conditions that define oriented, framed Legendrian links. Determining when two oriented, framed Legendrian links are equivalent or distinct is a classically difficult problem.
The primary goal of this dissertation is to develop systematic methods for distinguishing oriented, framed Legendrian links. As a first step, we describe how these topological objects, two-dimensional ribbons embedded in three-dimensional space, can be represented by one-dimensional diagrams in the plane. These planar representations which we call framed front diagrams, encode all of the relevant information of the oriented, framed Legendrian link, and our constructions operate entirely with these diagrams.
Our systematic methods uses ideas from category theory. A category consists of a collection of objects together with morphisms (maps) between them that satisfy certain structural properties. For suitable categories C, we construct a procedure that associates to each framed front diagram representing an oriented, framed Legendrian link a corresponding morphism in C. This assignment does not change when using different framed front diagrams that represent equivalent oriented, framed Legendrian links. As a result, if two oriented, framed Legendrian links are associated to different morphisms in C, we can conclude that the oriented, framed Legendrian links are distinct.
- Academic Unit
- Mathematics
- Record Identifier
- 9985177270402771
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