A combinatorial approach to knot theory : volume bounds for hyperbolic semi-adequate link complements
Giambrone, Adam Joseph
Knot theory
Combinatorial analysis
Geometry, Hyperbolic
Graph theory
Mathematics
Thesis Ph. D. Michigan State University. Mathematics - Doctor of Philosophy 2014.
An interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed braids, and families of links that have both plat and closed braid aspects. By more closely studying each of these families, we can often improve the lower bounds on volume provided by the main result. Furthermore, we show that the bounds on volume can be expressed in terms of a single stable coefficient of the colored Jones polynomial. By doing this, we provide new collections of links that satisfy a Coarse Volume Conjecture. The main approach of this entire work is to use a combinatorial perspective to study the connections among knot theory, hyperbolic geometry, and graph theory.
Includes bibliographical references.
Description based on online resource; title from PDF t.p. (viewed on May 4, 2017)
Kalfagianni, Efstratia
Kalfagianni, Efstratia
Hedden, Matthew
Gerhardt, Teena
Bell, Robert
Schmidt, Benjamin
2014
text
Electronic dissertations
Academic theses
application/pdf
1 online resource (xi, 132 pages) : illustrations
isbn:9781321060508
isbn:1321060505
umi:3629288
local:Giambrone_grad.msu_0128D_13004
en
In Copyright
Ph.D.
Doctoral
Mathematics - Doctor of Philosophy
Michigan State University