Casson towers and filtrations of the smooth knot concordance group

The 4-dimensional equivalence relation of concordance (smooth or topological) gives a group structure on the set of knots, under the connected-sum operation. The n-solvable filtration of the knot concordance group (denoted C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric attributes of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of C due to Cochran-Harvey-Horn, the positive and negative filtrations. The positive and negative filtrations have definitions similar to that of the n-solvable filtration, but have the ability (unlike the n-solvable filtrations) to distinguish between smooth and topological concordance. Our geometric counterparts for the positive and negative filtrations of C are defined in terms of Casson towers, 4-dimensional objects which approximate disks in a precise manner. We establish several relationships between these new Casson tower filtrations and the various previously known filtrations of C, such as the n-solvable, positive, negative, and grope filtrations. These relationships allow us to draw connections between some well-known open questions in the field.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

2014

Author dc:creator

• Ray, Arunima

Advisor dc:contributor.advisor

• Cochran, Tim D.

Committee members dc:contributor.committeemember

• Harvey, Shelly
• Warren, Joe

• Knot theory
• Knot concordance
• Casson towers

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/77507

OAI identifier oai:identifier

oai:repository.rice.edu:1911/77507