Title

Combinatorics of Twist Boxes of Knots

Faculty Sponsor

Paul Drube

College

Arts and Sciences

Department/Program

Mathematics and Statistics

ORCID Identifier(s)

0000-0003-2679-9327

Document Type

Poster Presentation

Symposium Date

Spring 4-23-2016

Abstract

In the study of knots, we can look at twist boxes--a set of consecutive half-twists in a knot diagram. Our project considers generalizations of twist boxes to higher-order twist boxes. We seek what combinations of first- and second-degree twist boxes exist in knots and how to construct any knot with those arbitrary combinations. We carefully define higher-order twist boxes in terms of checkerboard graphs of knots, and then confirm that the number of higher-order twist boxes depends only on the underlying knot chosen, not the graph or diagram used to represent it. We found several general cases showing general relations of number of degree one and degree two twist boxes of a knot that can coexist. Past results have shown the number of twist boxes corresponds to coefficients in the Jones polynomial, and for the future, we will seek a connection between degree two twist boxes and the Jones polynomial.

Biographical Information about Author(s)

Nichole Smith entered Valpo as a mathematics major, eager to study pure mathematics. Thus far she's had the pleasure of upper-level courses in combinatorics, analysis, and experimental mathematics. When Professor Drube offer her the chance to do some research with him, she, of course, took the opportunity. The summer research then turned into a project during the year. In some of her free time, she runs the problem-solving club, doing assorted problems instead of assigned ones.

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