Christine Caples
# 2017

Department of Applied Mathematical and Computational Sciences, The University of Iowa, Iowa City, IA 52242, UNITED STATES OF AMERICA.

### ABSTRACT

An *n*-string tangle is a three-dimensional ball with n strings properly embedded in it. Tangles have been used to model various biological processes involving DNA such as recombination. This thesis has two parts:*U* as those tangles obtained from *U* by allowing the boundary of *U* to move. For example, the family of a zero-crossing tangle is the set of rational tangles. An invariant is a quantity that is the same for equivalent tangles. We use the coloring invariant to classify the members of the family of an arbitrary tangle *U*.

We continue to improve upon an algorithm for tabulating tangles via crossing number using the Dowker notation for tangles. This algorithmic notation allows us to generate a list of all possible tangles of a particular crossing number. The algorithm removes many but not all tangles which can undergo simplification moves. Finally we explore the idea of using tangle families for tangle tabulation.

- To introduce the notion of tangle families and classify the family of an arbitrary tangle
*U*. - To further develop a computational algorithm for tabulating tangles via crossing number.

We continue to improve upon an algorithm for tabulating tangles via crossing number using the Dowker notation for tangles. This algorithmic notation allows us to generate a list of all possible tangles of a particular crossing number. The algorithm removes many but not all tangles which can undergo simplification moves. Finally we explore the idea of using tangle families for tangle tabulation.