DOI

10.17077/etd.fhxo-9mn8

Document Type

Dissertation

Date of Degree

Fall 2018

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Kawamuro, Keiko

First Committee Member

Fang, Hao

Second Committee Member

Frohman, Charles

Third Committee Member

Cooper, Ben

Fourth Committee Member

Tomova, Maggy

Abstract

We investigate various forms of link positivity: braid positivity, strong quasipositivity, and quasi- positivity. On the one hand, this investigation is undertaken in the context of braid simplification: we give sufficient conditions under which a given braid word is conjugate to a braid word with strictly fewer negative bands. On the other hand, we use the famous Bennequin inequality to define a new link invariant: the defect of the Bennequin inequality, or 3-defect, and give criteria in terms of the 3-defect under which a given link is (strongly) quasipositive.

Moreover, we use the 4-dimensional analogue of the Bennequin inequality, the slice Bennequin inequality in order to define the analogous defect of the slice Bennequin inequality, or 4-defect. We then investigate the relationship between the 4-defect and the most complicated class of 3- braids, Xu’s NP-form 3-braids, and establish several bounds. We also conjecture a formula for the signature of NP-form 3-braids which uses a new and easily computable NP-form 3-braid invariant, the offset.

Finally, the appendices provide lists of all quasipositive and strongly quasipositive knots with at most 12 crossings (with two exceptions, 12n239 and 12n512), along with accompanying quasipositive or strongly quasipositive braid words. Many of these knots did not have previously established positivities or braid words reflecting these positivities—these facts were discovered using various criteria (conjectural or proven) expressed throughout this thesis.

Keywords

Bennequin Inequality, Braid Theory, Contact Geometry, Strongly Quasipositive, Topology

Pages

viii, 119 pages

Bibliography

Includes bibliographical references (pages 116-119).

Copyright

Copyright © 2018 Jesse A. Hamer

Included in

Mathematics Commons

Share

COinS