#### Title

Research on Farey Recursion

#### Presentation Type

Presentation

#### Abstract

Linear recursion, think Fibonacci numbers, can be thought of as recursion along a line. Farey recursion is a way to describe a collection of polynomials which are recursive on a tree instead. This is useful in applications to geometry, topology, and number theory. This is based on a famous triangulation of the plane called the Farey Graph. Our research has been done alongside Professor Eric Chesebro and Professor Kelly McKinnie with background gained from Allen Hatcher's book Topology of Numbers. Our presentation will describe the Farey Graph and its close cousin, the Stern-Brocot Diagram, which we use to make the Farey recursion definition. Some patterns we have found using computer programs revolve around a family of polynomials T(p/q), which are indexed by the rational numbers. We will describe these patterns and will prove they lead to a closed formula for the polynomials that correspond to numbers of the form 1 /n. Our presentation will finish with a discussion of the questions still being examined in our ongoing work. We hope to contribute to the field of topology with this research by connecting Farey Recursion to other known aspects of this field including Knot Theory and Jones polynomials.

#### Category

Physical Sciences

Research on Farey Recursion

UC 327

Linear recursion, think Fibonacci numbers, can be thought of as recursion along a line. Farey recursion is a way to describe a collection of polynomials which are recursive on a tree instead. This is useful in applications to geometry, topology, and number theory. This is based on a famous triangulation of the plane called the Farey Graph. Our research has been done alongside Professor Eric Chesebro and Professor Kelly McKinnie with background gained from Allen Hatcher's book Topology of Numbers. Our presentation will describe the Farey Graph and its close cousin, the Stern-Brocot Diagram, which we use to make the Farey recursion definition. Some patterns we have found using computer programs revolve around a family of polynomials T(p/q), which are indexed by the rational numbers. We will describe these patterns and will prove they lead to a closed formula for the polynomials that correspond to numbers of the form 1 /n. Our presentation will finish with a discussion of the questions still being examined in our ongoing work. We hope to contribute to the field of topology with this research by connecting Farey Recursion to other known aspects of this field including Knot Theory and Jones polynomials.