Year of Award

2026

Document Type

Dissertation

Degree Type

Doctor of Philosophy (PhD)

Degree Name

Mathematics

Department or School/College

Department of Mathematical Sciences

Committee Chair

Kelly McKinnie

Commitee Members

Eric Chesebro, Nikolaus Vonessen, Cory Palmer, Lucia Williams

Abstract

The Stern-Brocot diagram is the graph whose vertices are the points (p/q, 1/q) where q > 0 and gcd(p, q) = 1, together with a point at infinity denoted by 1/0. Two vertices p/q and r/s are connected by an edge exactly when |ps - qr| = 1. A Farey recursive function assigns values in a commutative ring to these vertices so that, along any line formed by edges in the diagram, the values satisfy a second-order linear recurrence. Although these functions arise naturally in hyperbolic geometry, 2-bridge links, and Dehn filling constructions, this dissertation studies their algebraic properties.

First, we study values of Farey recursive functions at Stern-Brocot vertices lying on Euclidean lines with rational x-intercepts. We show that the corresponding values satisfy linear recurrence relations even when the vertices are not connected by edges. We call these new relations cross recursion. Second, for integer-valued Farey recursive functions, we study the periods of the resulting recursive sequences modulo a prime. We develop an algorithm for finding the period of a linear recursive sequence of arbitrary order modulo a prime and apply it to sequences arising from cross recursion.

For polynomial-valued Farey recursive functions, we study limits of zeros of recursively defined polynomial sequences. Using the Beraha-Kahane-Weiss theorem, we compare the nonisolated limits of zeros along Euclidean lines with those along lines in the Stern-Brocot diagram. Finally, we investigate discriminant roots associated with the characteristic polynomial of the second-order recurrence. For the Q-polynomials (defined by a particular Farey recursive function arising from 2-bridge link geometry), we show that, under a uniqueness hypothesis, roots of least modulus along a Euclidean line converge to the discriminant root of least modulus. We also examine how different algebraic rules for selecting roots can lead to convergence to different discriminant roots.

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