#### Presentation Title

Proof of Suzuki's Epimorphism Number

#### Faculty Mentor

Patrick Shanahan, Blake Mellor

#### Start Date

18-11-2017 10:00 AM

#### End Date

18-11-2017 11:00 AM

#### Location

BSC-Ursa Minor 113

#### Session

Poster 1

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

Author: Josh Ocana

Advisors: Dr. Blake Mellor, Dr. Patrick Shanahan

Take an extremely thin piece of string, tangle it up and then connect the ends to form a “knotted” loop in space. Informally, this is how we define a *knot* in mathematics, and the study of these objects is called knot theory. While knot theory has applications to chemistry and physics, this research project aims to further understand contemporary theoretical results regarding a specific relationship between a class of knots called 2-bridge knots. In 2017, the mathematician Masaaki Suzuki defined the *epimorphism number,* *EK(n)*, as the maximum number of 2-bridge knots a 2-bridge knot with *n *crossings can map onto via a construction of Ohtsuki, Riley, and Sakuma. In his research article, Suzuki established several theoretical results about *EK(n)* and used a computer program to compute *EK(n)* for *3 ≤ *n ≤ 30. He also posed the question of whether *EK(n) ≥ 3* for *31 ≤ n ≤ 44*. This project aims to answer said question by proving that *EK(n) ≤ 2* for *31 ≤ n ≤ 44* . We establish this result using techniques developed by Garrabrant, Hoste, and Shanahan associated to parsings of a continued fraction of the 2-bridge knot. Ultimately, the results of this research introduce a method to theoretically bound *EK(n)* that do not rely on an exhaustive computation by computer program.

**Keywords**: knot, 2-brige knot, crossings, epimorphism number, parsing.

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Proof of Suzuki's Epimorphism Number

BSC-Ursa Minor 113

Author: Josh Ocana

Advisors: Dr. Blake Mellor, Dr. Patrick Shanahan

Take an extremely thin piece of string, tangle it up and then connect the ends to form a “knotted” loop in space. Informally, this is how we define a *knot* in mathematics, and the study of these objects is called knot theory. While knot theory has applications to chemistry and physics, this research project aims to further understand contemporary theoretical results regarding a specific relationship between a class of knots called 2-bridge knots. In 2017, the mathematician Masaaki Suzuki defined the *epimorphism number,* *EK(n)*, as the maximum number of 2-bridge knots a 2-bridge knot with *n *crossings can map onto via a construction of Ohtsuki, Riley, and Sakuma. In his research article, Suzuki established several theoretical results about *EK(n)* and used a computer program to compute *EK(n)* for *3 ≤ *n ≤ 30. He also posed the question of whether *EK(n) ≥ 3* for *31 ≤ n ≤ 44*. This project aims to answer said question by proving that *EK(n) ≤ 2* for *31 ≤ n ≤ 44* . We establish this result using techniques developed by Garrabrant, Hoste, and Shanahan associated to parsings of a continued fraction of the 2-bridge knot. Ultimately, the results of this research introduce a method to theoretically bound *EK(n)* that do not rely on an exhaustive computation by computer program.

**Keywords**: knot, 2-brige knot, crossings, epimorphism number, parsing.