# Matt Rathbun

### Assistant Professor

Mathematics

California State University, Fullerton

McCarthy Hall 160B

mrathbun(at)fullerton(dot)edu

Office phone: 657-278-7697

Mathematics

California State University, Fullerton

McCarthy Hall 160B

mrathbun(at)fullerton(dot)edu

Office phone: 657-278-7697

I study *Knot* Theory and *3-manifolds*.

My research involves fibered knots, plumbing operations, Seifert surfaces, unknotting tunnels, Heegaard splittings, and applied knot theory to model DNA-protein interactions.

A knot is exactly what you would think - take a piece of string and tie it up. Only, instead of leaving the ends free, fuse them together, so that you cannot undo the knot. Easy to describe. But the same knot can take on many configurations in space, and things become complicated quite quickly. Suppose I tie two knots, behind my back, hand them to you, and ask you to decide whether they are the same, or different. You will soon find that fancifully trying to rearrange them to look identical is not efficient, or even guaranteed to provide an answer. We need a more systematic way of analyzing these objects.One rich method of studying knots is thinking about the space around the knot, but excluding the knot itself. This is but one example of a 3-manfiold. A 3-manifold "looks" like ordinary three-dimensional space, but more interesting phenomena can occur. For instance, every lasso can be tightened in ordinary space. But if you exclude a knot from space, then a lasso going around that knot would get "caught", and could not tighten any further. There are even three-manifolds where if you traveled in a rocketship in a straight line for long enough, you would eventually return right to your starting point, but as a mirror image of yourself!

These topics are fascinating, and fun, and also have applications for understanding the configurations of our DNA!

I am working on several projects with Ken Baker, Jessica Banks , Dorothy Buck, Marion Moore Campisi, Alexander Coward, Nicholas Jackson, Colin Johnson, Kai Ishihara, Koya Shimokawa , and Mike Williams.

*Monodromy action on unknotting tunnels in fiber surfaces*(with Jessica Banks)*The complete characterisation of efficient decatenation by the FtsK-dependent XerCD Recombinase*(with Dorothy Buck, Kai Ishihara, and Koya Shimokawa)*Band surgeries and crossing changes between fibered links*(with Dorothy Buck, Kai Ishihara, & Koya Shimokawa) [arXiv]*Tunnel one, fibered links*[arXiv]*Pacific Journal of Mathematics*, 259-2 (2012), 473--481. DOI 10.2140/pjm.2012.259.473*High distance knots in closed 3-manifolds*(with Marion Moore Campisi) [arXiv]*The visual boundary of Z^2*(joint with Kyle Kitzmiller) [arXiv]*Surfaces of high topological index in hyperbolic 3-manifolds*(with Marion Moore Campisi)

submitted

[preprint available by request]

submitted.

*Journal of Knot Theory and its Ramifications*, Vol 21, No 2 (2012)

Involve, a Journal of Mathematics, Vol 4, No 2 (2011), 103--116.

(This is work that resulted from an REU I helped supervise in 2006, sponsored and headed by Moon Duchin.)

[in preparation]

- Math 307 - Linear Algebra - Spring, 2015
- Math 414 - Topology - Fall, 2014
- Math 150A - Calculus I - Fall, 2014
- Bespoke Mathematics: Algebra, Analysis, Linear Algebra - Spring 2014
- Biological Mathematics - Autumn 2013
- Biological Mathematics - Autumn 2012
- Math 330 - Higher Geometry - Summer, 2011
- Math 432 - Axiomatic Geometry - Spring, 2011
- Math 310 - Abstract Algebra - Fall, 2010
- Math 108 - Intro to Abstract Math - Summer, 2010
- Calculus 21C - Summer, 2008
- Calculus 21A - Winter, 2006
- Calculus 21C - Winter, 2006
- Calculus 21A - Fall, 2006

- CSUF Geometry and Topology Seminar - 2014 - 2015 Organizer
- Imperial Festival - 2013
- London Topology and Geometry Seminar - 2012-2013 Co-Organizer
- SpartaNature - Module on Knot Theory - Summer 2011
- COSMOS - Problem Solving Strategies - Summer 2009
- Conference on 3- and 4-Dimensional Topology: In Honor of Martin Scharlemann, Davis, CA; June 23-25 2009 Local Organizing Committee
- Junior Topologists' Research Retreat, Davis, CA; June 26-29 2009 Co-Organizer