On the Breadth of the Jones Polynomial for Certain Classes of Knots and Links

The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to find closed formulas for the smallest and largest exponents of the Jones polynomial for certain classes of knots and links. This allows us to find a lower bound on the crossing number for these knots and links very quickly. These formulas for the smallest and largest exponents of the Jones polynomial are constructed from special rational tangles expansions and using these formulas, we can extend these results to for [sic] special cases of Montesinos knots and links.

Name thesis:degree_name

Master of Science

Discipline thesis:degree_discipline

Department of Mathematics and Computer Science

Year

2009

Author dc:creator

• Lorton, Cody

Contributors dc:contributor

• Dr. Claus Ernst (Director), Dr. Mark P. Robinson,Dr. Uta Ziegler

• knot theory
• Montesinos knots
• Jones polynomial
• Mathematics
• Number Theory

Repository record dc:identifier

https://digitalcommons.wku.edu/theses/86

OAI identifier oai:identifier

oai:digitalcommons.wku.edu:theses-1081