Dissertations, Theses, and Capstone Projects

Date of Degree

6-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Ilya Kofman

Committee Members

Abhijit Champanerkar

Joshua Sussan

Adam Lowrance

Subject Categories

Geometry and Topology

Keywords

Turaev surfaces, Toroidally alternating knots

Abstract

In this thesis, we study knots and links via their alternating diagrams on closed orientable surfaces. Every knot or link has such a diagram by a construction of Turaev, which is called the Turaev surface of the link. Links that have an alternating diagram on a torus were defined by Adams as toroidally alternating. For a toroidally alternating link, the minimal genus of its Turaev surface may be greater than one. Hence, these surfaces provide different topological measures of how far a link is from being alternating.

First, we classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating.

Second, we give a topological characterization of toroidally alternating knots and almost-alternating knots. In other words, we provide necessary and sufficient topological conditions for a knot to be toroidally alternating or almost-alternating. Our topological characterization extends Howie’s characterization of alternating knots, but is different from Ito’s characterization of almost-alternating knots.

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