Date of Award
Master of Arts in Mathematics
First Reader/Committee Chair
A knot is a closed curve in R3. Alternatively, we say that a knot is an embedding f : S1 → R3 of a circle into R3. Analogously, one can think of a knot as a segment of string in a three-dimensional space that has been knotted together in some way, with the ends of the string then joined together to form a knotted loop. A link is a collection of knots that have been linked together.
An important question in the mathematical study of knot theory is that of how we can tell when two knots are, or are knot, equivalent. That is, we would like to know when two seemingly different looking knots are, or are not, in fact the same knot. In this thesis, we discuss knots, links, and the concepts necessary for establishing knot equivalence.
We seek to show when two knots are equivalent through a process known as ambient isotopy. We seek to show when two knots are not equivalent through a process known as knot invariance. We will discuss special types of isotopies, such as Reidemeister moves, and we will cover various types of invariants, such as linking number, tricolorability, and the Jones polynomial.
Trubey, Jacob, "KNOT EQUIVALENCE" (2023). Electronic Theses, Projects, and Dissertations. 1631.