#### Date of Award

2017

#### Document Type

Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Dr. Heather Russell

#### Abstract

Knot theory arguably holds claim to the title of the mathematical discipline with the most unusually diverse applications. A *knot *can be defined topologically as an embedding of *S*1 in R3. Naturally, two knots are topologically equivalent if one cannot be smoothly deformed into the other. The question of whether two knots are equivalent is highly non-trivial, and so the question of knot invariants used to distinguish knots has occupied knot theorists for over a century. Knot theory has found application in statistical mechanics [1], symbolic logic and set theory [2], quantum fi theory [3], quantum computing [4], etc. This thesis focuses on a connection of knot invariants to a still evolving fi quantum groups. The representation theory of a particular quantum group, *U**q *psl2pCqq , encodes information that, when expressed via a knot diagram in a well-defined graphical calculus, produces the Jones polynomial, arguably the most famous of knot invariants. Section 1 gives an introduction of this quantum group. Section 2 details the representation theory of *U**q *psl2pCqq . Section 3 introduces category theory and the category *Rep**U**q *, and shows how *Rep**U**q *can produce the Jones polynomial through an example with the trefoil knot.

#### Recommended Citation

Hamilton, Greg A., "Quantum Groups and Knot Invariants" (2017). *Honors Theses*. 975.

https://scholarship.richmond.edu/honors-theses/975