Title page for ETD etd-07032012-165733

Type of Document Dissertation
Author Armond, Cody
URN etd-07032012-165733
Title The Head and Tail Conjecture for Alternating Knots
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Dasbach, Oliver Committee Chair
Davidson, Mark Committee Member
Gilmer, Patrick Committee Member
Hoffman, Jerome Committee Member
Litherland, Richard Committee Member
Peng, Lu Dean's Representative
  • knot theory
  • colored Jones polynomial
  • skein theory
  • head and tail
  • adequate links
Date of Defense 2012-03-27
Availability unrestricted
The colored Jones polynomial is an invariant of knots and links, which produces a sequence of Laurent polynomials. In this work, we study new power series link invariants, derived from the colored Jones polynomial, called its head and tail. We begin with a brief survey of knot theory and the colored Jones polynomial in particular. In Chapter 3, we use skein theory to prove that for adequate links, the n-th leading coefficient of the N-th colored Jones polynomial stabilizes when viewed as a sequence in N. This property allows us to define the head and tail for adequate links. In Chapter 4 we show a class of knots with trivial tail, and in Chapter 5 we develop techniques to calculate the head and tail for various knots and links using a graph derived from the link diagram.
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