#### Document Type

Dissertation

#### Date of Degree

Spring 2016

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Charles Frohman

#### Abstract

There is an algebra defined on a two dimensional manifold, known as the Skein algebra, which has as elements the simple closed curves of the manifold. Just like with numbers, there's a way to add, subtract and multiply elements. Unfortunately division is not allowed in the Skein algebra, which is why we introduced the notion of the Localized Skein Algebra, where we define a way to invert elements so that dividing is possible. These algebras have infinitely many elements, may not be commutative and in fact may have torsion, which makes them a hard object to study.

This work is mainly centered in reducing these algebras to something more manageable. We have shown that for any space, its Localized Skein Algebra is a Frobenius extension of its Localized Character Ring, which means that any element of the algebra can be rewritten as a finite linear combination of a finite subset of basis elements, multiplied by elements that do commute. The importance of this result is that it solves this problem of noncommutativity, by rewriting anything that doesn't commute, as elements of a small set which can be controlled, along with elements that commute and behave nicely, making the Skein algebra far more manageable.

#### Public Abstract

Take two of your friends, and if you do not have any friends, take two of your Facebook friends. Give each one of them a piece of string and have them tie as many knots as they want; after they are done, have them glue both ends of the string. The resulting object is what we, mathematicians, call a knot. Does there exists a computational algorithm that can compare whether your friends ended up with the same knot or not? The answer is: Yes, and these are called in mathematics knot invariants. It was previously thought that computing with these algorithms required exponential time. The importance of this research and its results is that shows that there is a rare subset of knots that allows you to compute in linear time instead. We proved this to be the case for any type of two dimensional space.

#### Keywords

publicabstract, Algebra, Skein, Topology

#### Pages

vi, 80 pages

#### Bibliography

Includes bibliographical references (pages 78-80).

#### Copyright

Copyright 2016 Nelson Abdiel Colon