I study knots and links, as well as the 3-dimensional manifolds that surround them. I was first drawn to this field by its visual nature and strong geometric flavor. I like seeing things visually, and the pictures of hyperbolic geometry play a significant role in my work. Here is what I do, in a little more detail: A description of my work in plain English (with thanks to Stanford's I-RITE Program) My research statement (PDF), as of Fall 2007. Curriculum vitae (PDF), as of Fall 2008.

10. Cusp areas of Farey manifolds and applications to knot theory. With Effie Kalfagianni and Jessica Purcell.
    Submitted (2008). [PS], [PDF], [ArXiv].

9. Symmetric links and Conway sums: volume and Jones polynomial. With Effie Kalfagianni and Jessica Purcell.
   Submitted (2008). [PS], [PDF], [ArXiv].

8. Alternating sum formulae for the determinant and other link invariants. With Oliver Dasbach, Effie Kalfagianni, Xiao-Song Lin, and Neal Stoltzfus.
   Submitted (2007). [PS], [PDF], [ArXiv].

7. The Jones polynomial and graphs on surfaces. With Oliver Dasbach, Effie Kalfagianni, Xiao-Song Lin, and Neal Stoltzfus.
   Journal of Combinatorial Theory, Series B 98 (2008), Issue 2, 384-399. [PDF], [Web], [ArXiv].

6. Dehn filling, volume, and the Jones polynomial. With Effie Kalfagianni and Jessica Purcell.
   Journal of Differential Geometry 78 (2008) 429-464. [PDF], [Web], [ArXiv].

5. Angled decompositions of arborescent link complements. With François Guéritaud.
   To appear in Proceedings of the London Mathematical Society (2008). [PDF], [Web], [ArXiv].

4. Geometric triangulations of two-bridge link complements. Appendix to a paper by François Guéritaud.
   Geometry & Topology 10 (2006), 1267-1282. [PDF], [Web], [ArXiv].

3. Links with no exceptional surgeries. With Jessica Purcell.
   Commentarii Mathematici Helvetici 82 (2007), No. 3, 629-664. [PDF], [Web], [ArXiv].

2. Involutions of knots that fix unknotting tunnels.
   Journal of Knot Theory and its Ramifications 16 (2007), No. 6, 741-748. [PDF], [Web], [ArXiv].

1. Cost-minimizing networks among immiscible fluids in R². With Andrei Gnepp, David McMath, Brian Munson, Ting Ng, Sang-Hyoun Pahk, and Cara Yoder.
   Pacific Journal of Mathematics 196 (2000), no. 2, 395-414. [PS], [PDF], [Web].

• William Thurston's Geometry and Topology of 3-Manifolds -- the bible of my field
• ArXiv Preprint server for papers in geometric topology
• MathSciNet
• The SnapPea program by Jeff Weeks, including Nathan Dunfield's additions
• Interactive knot tables, by Charles Livingston

Visualization software:

• The KnotPlot Site makes beautiful pictures, including both of the knot pictures on this page.
• Not Knot, a great visual introduction to hyperbolic geometry
• Spend an hour or two playing with Jeff Weeks's programs, and you'll understand the idea of a manifold.
• More from the Geometry Center (now sadly defunct)

• Homepages of low-dimensional topologists, courtesy of Mark Brittenham
• Low-dimensional topology conferences, also from Mark Brittenham
• The Math Genealogy Project. Believe it or not, I am 14 generations down from Euler.
• My immediate mathematical family