#### Title

Geometry of an (infinite) family of tangles

#### Presentation Type

Presentation

#### Faculty Mentor’s Full Name

Eric Chesebro

#### Faculty Mentor’s Department

Mathematical Sciences

#### Abstract

Topology is a branch of mathematics which may be loosely described as the study of "flexible" spaces. While fascinating from a theoretical standpoint, topology also has many applications in such diverse fields as genetics, superconductors and robotics. One particularly interesting branch of topology is the study of knots, and their cousins, links, and tangles. A knot can be visualized as a loop of string in 3-space. A tangle is similar, this time using strings, which are not joined up into a loop, embedded in a solid ball so that only the ends of the strings lie on the boundary of the ball. Because of their geometric properties, we are especially interested in a class of 2-strand tangles called irrational tangles. A very powerful tool for understanding and distinguishing tangles is hyperbolic geometry. In comparison to Euclidean geometry, hyperbolic geometry is where triangles are "thin," meaning that the interior angles sum to less than 180 degrees, and parallel lines are not unique. We are working on how to understand the geometry of a certain infinite family of irrational tangles. We are examining the less complex members of the family in order to generalize the geometry for all members.

#### Category

Physical Sciences

Geometry of an (infinite) family of tangles

UC 326

Topology is a branch of mathematics which may be loosely described as the study of "flexible" spaces. While fascinating from a theoretical standpoint, topology also has many applications in such diverse fields as genetics, superconductors and robotics. One particularly interesting branch of topology is the study of knots, and their cousins, links, and tangles. A knot can be visualized as a loop of string in 3-space. A tangle is similar, this time using strings, which are not joined up into a loop, embedded in a solid ball so that only the ends of the strings lie on the boundary of the ball. Because of their geometric properties, we are especially interested in a class of 2-strand tangles called irrational tangles. A very powerful tool for understanding and distinguishing tangles is hyperbolic geometry. In comparison to Euclidean geometry, hyperbolic geometry is where triangles are "thin," meaning that the interior angles sum to less than 180 degrees, and parallel lines are not unique. We are working on how to understand the geometry of a certain infinite family of irrational tangles. We are examining the less complex members of the family in order to generalize the geometry for all members.