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Research

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Courses

    Spring 2009
    CTI 104
    MAT 200
    MAT 281
    MAT 451

    Fall 2009
    CTI 103
    CTI 104
    MAT 201
    MAT 320

Research

Current Work (Personal)

Specifically, I am interested in the relationships between different knot invariants.  My first paper investigates the effect numerous knot invariants have on properties of the Jones polynomial.  While working on this, I became familiar with virtual knots and flat virtual knots.   I have classified up to six-edged PL-virtual knots and am investigating the remaining knots in the virtual knot table. 

Lastly, I have been interested for some time in the idea of resolving 4-valent planar graphs to both classical and virtual knot diagrams (also known as pseudodiagram theory).  This is a fairly straighforward concept, and if you are an undergraduate reading this and interested in exploring the concept further, don't hesitate to contact me!

Current Work (Undergraduate Research)
Chris Sharp (now a student at the University of Kansas) and I worked on original research on the visibility of lattice points.  The idea came to us from a wonderful paper in the MAA's Mathematics Magazine by Josh Laison (whom we are grateful for his helpful comments) and Michelle Schick.  Our paper has been submitted.

A second project titled Age, gender, and knot tying took place over the 08/09 academic year with Elizabeth Waterland.  Inspired by an article in Physics Today that investigated tumbling string in a box and investigating what knots were formed, we are providing a brief (a minute or two) introduction to knots to persons of all ages and education levels and then asking them to tie knots.  Using various knot theoretic techniques, we determine what knots were formed.  Eventually, we aim to answer if age, gender, or education level influence "how complex" of a knot a person ties.  Elizabeth is presented her results at the 2009 William Jewell College Duke Colloquium Day, and our paper is currently under review.

Future Work
I plan to continue investigating the o-length invariant defined on knots and further consider its relationships with other knot invariants.  Beyond this, I am extremely interested in knots and links in projective 3-space, the space of piecewise-linear knots (see my preprint at the arXiv below), virtual knot theory, and connections of my work with graph theory.  The work with Chris Sharp has opened my eyes to many questions in lattice point theory, and I plan to continue these investigations with future students.

Publications
On Knots and Their Invariants, Ph.D. dissertation, University of Iowa, 2007
Nonalternating knots and Jones polynomials, J. Knot Theory Ramifications, 17(8): 983-1003, 2008.
Weakly viewing lattice points, with R. Christopher Sharp, submitted
Universe resolutions and bridge number, submitted
Piecewise-linear virtual knots, in preparation, arXiv:math/09072215
Age, gender and knot tying, with Elizabeth Waterland, in preparation

Selected Presentations
Church, Chess, and Knots: Statistics Through Service and Research, MAA General Session, Joint Mathematics Meetings, San Francisco, Jan. 2010
Resolving Knot Universes, MAA-Missouri Section Meeting, Truman State University, April 2009
Strengthening Kauffman-Murasugi via a Skeleton, University of Kansas Combinatorics Seminar, Nov. 2008
On the Visibility of Lattice Points, UMKC Mathematics Talk Series, University of Missouri-KC, Nov. 2008
A polynomial for flat virtual knots, AMS Session, Geometry and Topology, Joint Math. Meetings, San Diego, Jan. 2008
Knots, Virtually Speaking, UMKC Mathematics Talk Series, University of Missouri-KC, Nov. 2007
Tied Up In Knots: An Introduction to Mathematical Knots, KME/SPS, William Jewell College, Oct. 2007
1, 2, 3, 4: Counting Problems Galore!, Undergraduate Research Seminar, The University of Iowa, March 2007
Virtual knots: x marks the spot, Bradley University, February 2007
Knots & Their Invariants, Lake Forest College, February 2007
Jaeger's Model for the HOMFLY Polynomial, VIGRE Topology Seminar, The University of Iowa, January 2007
Nonalternating knots and the Jones polynomial, Invited Special Session talk, AMS National Meeting, New Orleans, January 2007
Persistent homology, VIGRE Topology Seminar, The University of Iowa, September 2006
Nonalternating knots and Jones polynomials, Topology Seminar, The University of Iowa, September 2006
An ice model for the Jones polynomial, Graduate Student Topology Conference, Indiana University, April 2006
 Nonalternating skeletons, Topology Seminar, The University of Iowa, March 2006
Knot Invariant or Not?, Wartburg College (with Kathleen Reif), November 2005
An ice model for the Jones polynomial, Topology Seminar, The University of Iowa, October 2005
A polynomial invariant for knots via the Kauffman bracket, Topology Seminar, The University of Iowa, April 2005

 Please feel free to email me with specific questions or comments.

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