Date of Award
Bachelor of Science
Dr. Heather Russell
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 S1 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 , symbolic logic and set theory , quantum fi theory , quantum computing , 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, Uq 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 Uq psl2pCqq . Section 3 introduces category theory and the category RepUq , and shows how RepUq can produce the Jones polynomial through an example with the trefoil knot.
Hamilton, Greg A., "Quantum Groups and Knot Invariants" (2017). Honors Theses. 975.