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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Nov 18th, 10:00 AM Nov 18th, 11:00 AM

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.