## Theses and Dissertations

#### Title

The Khovanov homology of the jumping jack

#### DOI

10.17077/etd.gnhgj0wg

Dissertation

Spring 2015

#### Degree Name

PhD (Doctor of Philosophy)

Mathematics

Frohman, Charles

Cooper, Benjamin

Kawamuro, Keiko

Kinser, Ryan

Tomova, Maggy

#### Abstract

We study the sl(3) web algebra via morphisms on foams. A pre-foam is a cobordism between two webs that contains singular arcs, which are sets of points whose neighborhoods are homeomorphic to the cross-product of the letter "Y'' and the unit interval. Pre-foams may have a distinguished point, and it can be moved around as long as it does not cross a singular arc. A foam is an isotopy class of pre-foams modulo a set of certain relations involving dots on the pre-foams. Composition in Foams is achieved by stacking pre-foams. We compute the cohomology ring of the sl(3) web algebra and apply a functor from the cohomology ring of the sl(3) web algebra to {\bf Foams}. Afterwards, we use this to study the $\mathfrak{sl}(3)$ web algebra via morphisms on foams.

#### Public Abstract

New results in Quantum Physics led to the development of quantum computers, which are the next generation of supercomputers. They can work faster and more efficiently than current computers. This leads to problems in encryption because quantum computers can easily break all current forms of data encryption currently in use. The study of encryption methods for quantum computers is Quantum Encryption. Through the use of a mathematical process known as categorification, we study polynomials through a diagrammatic algebra known as Khovanov Homology. The results obtained here lead to new insights in the mathematical problems that arise in Quantum Encryption.

#### Keywords

publicabstract, Jones polynomial, Khovanov homology, knot, quantum, topology, webs

xii, 100 pages

#### Bibliography

Includes bibliographical references (page 100).