On the Number of Colors in Quandle Knot Colorings

Author

Jeremy William Kerr

Graduation Year

2016

Document Type

Thesis

Degree

M.A.

Degree Name

Master of Arts (M.A.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Mohamed Elhamdadi, Ph.D.

Committee Member

Brian Curtin, Ph.D.

Committee Member

Masahiko Saito, Ph.D.

Keywords

Knot Theory, Fox Coloring, Linear Alexander Quandle Coloring, Minimal Coloring

Abstract

A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five colors. The ideas behind the reduction of colors in a Fox coloring is extended to knots colored by linear Alexander quandles. Thus, we prove that any knot colored by either the linear Alexander quandle Z₅[t]/(t − 2) or Z₅[t]/(t − 3) has a diagram using only four colors.

Scholar Commons Citation

Kerr, Jeremy William, "On the Number of Colors in Quandle Knot Colorings" (2016). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/6103