#### Identifier

etd-06222015-181044

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

The tail of a sequence {P_n(q)} of formal power series in Z[q^{-1}][[q]], if it exists, is the formal power series whose first $n$ coefficients agree up to a common sign with the first n coefficients of P_n. The colored Jones polynomial is link invariant that associates to every link in S^3 a sequence of Laurent polynomials. In the first part of this work we study the tail of the unreduced colored Jones polynomial of alternating links using the colored Kauffman skein relation. This gives a natural extension of a result by Kauffman, Murasugi, and Thistlethwaite regarding the highest and lowest coefficients of Jones polynomial of alternating links. Furthermore, we show that our approach gives a new and natural proof for the existence of the tail of the colored Jones polynomial of alternating links. In the second part of this work, we study the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. This can be considered as a generalization of the study of the tail of the colored Jones polynomial. We use local skein relations to understand and compute the tail of these graphs. Furthermore, we consider certain skein elements in the Kauffman bracket skein module of the disk with marked points on the boundary and we use these elements to compute the tail quantum spin networks. We also give product structures for the tail of such trivalent graphs. As an application of our work, we show that our skein theoretic techniques naturally lead to a proof for the Andrews-Gordon identities for the two variable Ramanujan theta function as well to corresponding new identities for the false theta function.

#### Date

2015

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Hajij, Mustafa, "Knots, Skein Theory and q-Series" (2015). *LSU Doctoral Dissertations*. 258.

https://digitalcommons.lsu.edu/gradschool_dissertations/258

#### Committee Chair

Dasbach, Oliver