Novel Characterizations of Knot Complexity
Authors:Joshua Henrich, David Hyde, Kenneth Millett, Eric Rawdon
Mentor:Kenneth Millett, Professor of Mathematics, University of California Santa Barbara
In recent years, the field of knot theory has entered into the spotlight of mathematics. This is in large part due to the fact that, in addition to their role as pure mathematical structures, knots have found numerous applications in the sciences. For example, one may use knot theory to analyze the structures of proteins and to gain greater insight into the mechanisms by which proteins fold or unfold. Key to understanding such applications of knot theory are ways to characterize the complexity of knots. In this work, we analyze graph-theoretic and algebraic properties of "knot fingerprints" in order to develop a systematic characterization of knot complexity. We present ways to use the adjacency matrix of an associated graph as a means of representing knot fingerprints. Furthermore, we draw from the concepts of the Cheeger constant and Menger’s theorems and interpret knot fingerprints in the context of a network or flow. With these analyses, we are able to identify quantities that successfully measure knot complexity. We demonstrate the effectiveness of our characterizations by applying them to knots of up to ten crossings as well as select composite knots. Finally, we suggest new avenues of research and potential applications.