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) Curriculum vitae (PDF), as of Fall 2011. Temple Geometry-Topology group Group blog Temple Geometry-Topology Seminar Penn Geometry-Topology Seminar Temple Mathematics Colloquium

22. Fiber detection for state surfaces.
    Submitted (2012). [PDF], [ArXiv].

21. Jones polynomials, volume, and essential knot surfaces: a survey. With Effie Kalfagianni and Jessica Purcell.
    Submitted (2011). [PDF], [ArXiv].

20. Cusp geometry of fibered 3-manifolds. With Saul Schleimer.
    Submitted (2011). [PDF], [ArXiv].

19. Guts of surfaces and the colored Jones polynomial. With Effie Kalfagianni and Jessica Purcell.
    Submitted (2011). [PDF], [ArXiv].

18. Dehn filling and the geometry of unknotting tunnels. With Daryl Cooper and Jessica Purcell.
    Submitted (2011). [PDF], [ArXiv].

17. Explicit angle structures for veering triangulations. With François Guéritaud.
    Submitted (2011). [PDF], [ArXiv].

16. Surface quotients of hyperbolic buildings. With Anne Thomas.
    International Mathematics Research Notices 2012, Issue 2, 437-477. [PDF], [Web], [ArXiv].

15. Volume bounds for generalized twisted torus links. With Abhijit Champanerkar, Ilya Kofman, Walter Neumann, and Jessica Purcell.
    Mathematical Research Letters, to appear (2012). [PDF], [ArXiv].

14. From angled triangulations to hyperbolic structures. With François Guéritaud.
    Contemporary Mathematics 541 (2011), 159-182. [PDF], [Web], [ArXiv].

13. Slopes and colored Jones polynomials of adequate knots. With Effie Kalfagianni and Jessica Purcell.
    Proceedings of the American Mathematical Society 139 (2011), Issue 5, 1889-1896. [PDF], [Web], [ArXiv].

12. On diagrammatic bounds of knot volumes and spectral invariants. With Effie Kalfagianni and Jessica Purcell.
    Geometriae Dedicata 147 (2010), 115-130. [PDF], [Web], [ArXiv].

11. Finite surgeries on three-tangle pretzel knots. With Masaharu Ishikawa, Yuichi Kabaya, Thomas Mattman, and Koya Shimokawa.
    Algebraic & Geometric Topology 9 (2009), 743-771. [PDF], [Web], [ArXiv].

10. Cusp areas of Farey manifolds and applications to knot theory. With Effie Kalfagianni and Jessica Purcell.
    International Mathematics Research Notices 2010, Issue 23, 4434-4497. [PDF], [Web], [ArXiv].

9. Symmetric links and Conway sums: volume and Jones polynomial. With Effie Kalfagianni and Jessica Purcell.
   Mathematical Research Letters 16 (2009), Issue 2, 233-253. [PDF], [Web], [ArXiv].

8. Alternating sum formulae for the determinant and other link invariants. With Oliver Dasbach, Effie Kalfagianni, Xiao-Song Lin, and Neal Stoltzfus.
   Journal of Knot Theory and its Ramifications 19 (2010), Issue 6, 765-782. [PDF], [Web], [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. Angled decompositions of arborescent link complements. With François Guéritaud.
   Proceedings of the London Mathematical Society 98 (2009), Issue 2, 325-364. [PDF], [Web], [ArXiv].

5. Dehn filling, volume, and the Jones polynomial. With Effie Kalfagianni and Jessica Purcell.
   Journal of Differential Geometry 78 (2008) 429-464. [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 Fai Ng, Sang-Hyoun Pahk, and Cara Yoder.
   Pacific Journal of Mathematics 196 (2000), no. 2, 395-414. [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
• SnapPy by Marc Culler and Nathan Dunfield is an updated version of Jeff Weeks's SnapPea
• 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.
• The program Curved Spaces lets you fly through a 3-manifold and see its intrinsic geometry.

• 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