A description of my work in plain English. This was written in 2005, as I was finishing graduate school, but still describes one major theme in my research. Research overview (PDF), as of Summer 2019. Curriculum vitae (PDF), as of Winter 2025. Temple Geometry-Topology group Temple Geometry-Topology Seminar Penn Geometry-Topology Seminar Temple Mathematics Colloquium

45. On fixed points of pseudo-Anosov maps.
    With Tarik Aougab and Sam Taylor.
    Preprint on the arXiv.

44. There are no periodic Wright maps.
    Preprint on the arXiv.

43. Arithmeticity and commensurability of links in thickened surfaces.
    With Rose Kaplan-Kelly.
    To appear in Algebraic & Geometric Topology. arXiv.

42. Double-Anonymous Peer Review in Mathematics: Implementation for American Mathematical Society Journals.
    With Dan Abramovich, Henry Cohn, and Robert Harington.
    Notices of the AMS 71 (2024), Issue 7, 1079-1081. HTML.

41. Excluding cosmetic surgeries on hyperbolic 3-manifolds.
    With Jessica Purcell and Saul Schleimer.
    Preprint on the arXiv.

40. Homotopy equivalent boundaries of cube complexes.
    With Talia Fernos and Mark Hagen.
    Geometriae Dedicata 218 (2024), article #33. arXiv.

39. Large volume fibred knots of fixed genus.
    With Ken Baker, Jessica Purcell, and Saul Schleimer.
    Mathematical Research Letters 31 (2024), Issue 5, 1305-1314. arXiv.

38. Cubulating random quotients of hyperbolic cubulated groups.
    With Daniel Wise.
    Transactions of the AMS, Series B 11 (2024), 622-666. arXiv.

37. Infinitely many virtual geometric triangulations.
    With Emily Hamilton and Neil Hoffman.
    Journal of Topology 15 (2022), Issue 4, 2352-2388. arXiv.

36. Effective drilling and filling of tame hyperbolic 3-manifolds.
    With Jessica Purcell and Saul Schleimer.
    Commentarii Mathematici Helvetici 97 (2022), Issue 3, 457-512. arXiv.

35. Effective bilipschitz bounds on drilling and filling.
    With Jessica Purcell and Saul Schleimer.
    Geometry & Topology 26 (2022), Issue 3, 1077-1188. arXiv.

34. Random veering triangulations are not geometric.
    With Sam Taylor and Will Worden.
    Groups, Geometry, and Dynamics 14 (2020), Issue 3, 1077-1126. arXiv.

33. Effective distance between nested Margulis tubes.
    With Jessica Purcell and Saul Schleimer.
    Transactions of the American Mathematical Society 372 (2019), Issue 6, 4211-4237. arXiv.

32. A survey of hyperbolic knot theory.
    With Effie Kalfagianni and Jessica Purcell.
    Knots, Low-Dimensional Topology and Applications, Springer Proceedings in Mathematics & Statistics, vol. 284 (2019), 1-30. arXiv.

31. Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds.
    With Daryl Cooper.
    Geometry & Topology 23 (2019), Issue 1, 241-298. arXiv.

30. Growth of quasiconvex subgroups.
    With François Dahmani and Daniel Wise.
    Mathematical Proceedings of the Cambridge Philosophical Society 167 (2019), Issue 3, 505-530. arXiv.

29. Spectrally similar incommensurable 3-manifolds.
    With Christian Millichap.
    Proceedings of the London Mathematical Society 115 (2017), Issue 2, 411-447. arXiv.

28. The lowest volume 3-orbifolds with high torsion.
    With Chris Atkinson.
    Transactions of the American Mathematical Society 369 (2017), Issue 8, 5809-5827. ArXiv.

27. Hyperbolic semi-adequate links.
    With Effie Kalfagianni and Jessica Purcell.
    Communications in Analysis & Geometry 23 (2015), Issue 5, 993-1030. ArXiv.

26. Essential surfaces in highly twisted link complements.
    With Ryan Blair and Maggy Tomova.
    Algebraic & Geometric Topology 15 (2015), Issue 3, 1501-1523. ArXiv.

25. Quasifuchsian state surfaces.
    With Effie Kalfagianni and Jessica Purcell.
    Transactions of the American Mathematical Society 366 (2014), Issue 8, 4323-4343. ArXiv.

24. Cusp geometry of fibered 3-manifolds.
    With Saul Schleimer.
    American Journal of Mathematics 136 (2014), Issue 2, 309-356. ArXiv.

23. Small volume link orbifolds.
    With Chris Atkinson.
    Mathematical Research Letters 20 (2013), Issue 6, 995-1016. ArXiv.

22. Jones polynomials, volume, and essential knot surfaces: a survey.
    With Effie Kalfagianni and Jessica Purcell.
    Proceedings of Knots in Poland III, Banach Center Publications 100 (2014), Issue 1, 51-77. ArXiv.

21. Fiber detection for state surfaces.
    Algebraic & Geometric Topology 13 (2013), Issue 5, 2799-2807. ArXiv.

20. Guts of surfaces and the colored Jones polynomial.
    With Effie Kalfagianni and Jessica Purcell.
    Monograph published in Lecture Notes in Mathematics (Springer), volume 2069 (2013). ArXiv.

19. Dehn filling and the geometry of unknotting tunnels.
    With Daryl Cooper and Jessica Purcell.
    Geometry & Topology 17 (2013), Issue 3, 1815-1876. ArXiv.

18. Explicit Dehn filling and Heegaard splittings.
    With Jessica Purcell.
    Communications in Analysis and Geometry 21 (2013), Issue 3, 625-650. ArXiv.

17. Explicit angle structures for veering triangulations.
    With François Guéritaud.
    Algebraic & Geometric Topology 13 (2013), Issue 1, 205-235. ArXiv.

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

15. Volume bounds for generalized twisted torus links.
    With Abhijit Champanerkar, Ilya Kofman, Walter Neumann, and Jessica Purcell.
    Mathematical Research Letters 18 (2011), Issue 6, 1097-1120. ArXiv.

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

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

11. Finite surgeries on three-tangle pretzel knots.
    With Masaharu Ishikawa, Yuichi Kabaya, Thomas Mattman, and Koya Shimokawa.
    Algebraic & Geometric Topology 9 (2009), Issue 2, 743-771. 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. 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. 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. 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. ArXiv.

6. Angled decompositions of arborescent link complements.
   With François Guéritaud.
   Proceedings of the London Mathematical Society 98 (2009), Issue 2, 325-364. ArXiv.

5. Dehn filling, volume, and the Jones polynomial.
   With Effie Kalfagianni and Jessica Purcell.
   Journal of Differential Geometry 78 (2008), Issue 3, 429-464. ArXiv.

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

3. Links with no exceptional surgeries.
   With Jessica Purcell.
   Commentarii Mathematici Helvetici 82 (2007), Issue 3, 629-664. ArXiv.

2. Involutions of knots that fix unknotting tunnels.
   Journal of Knot Theory and its Ramifications 16 (2007), Issue 6, 741-748. 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), Issue 2, 395-414.

• 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, Nathan Dunfield and Matthias Goerner 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.

• Low-dimensional topology blog
• Low-dimensional topology conferences
• The Math Genealogy Project. Believe it or not, I am 14 generations down from Euler.
• My immediate mathematical family