Research

Graduate students (current):

• Julia Courtney

• Sam Lewis-Monkman

• Pablo Padilla

Graduate students (past):

• Vince Longo (PhD earned 2021, co-advised with Mark Brittenham)

• Nicholas Meyer (PhD earned 2024)

• Jesse Moeller (PhD earned 2021, co-advised with Mark Brittenham)

• Marla Williams (PhD earned 2020, co-advised with Mark Brittenham)

• Ana Wright (PhD earned 2022, co-advised with Mark Brittenham)

Undergraduate research assistants (current):

• Caleb Kelly

• Nathan McCormick

Undergraduate research assistants (past):

• Gabriel Adams

• Derek Davies

• Lucy DePooter

• Hanna Hoffman

• Andrew Loken

• Kenji Nakagawa

• Mia Nguyen

• Rilee Potter

• Jack Rankin

• Ethan Romary

• Rebecca Sorsen

• Olivia Thiel

Preprints:

1. (with Trevor Olveira-Smith and Tye Lidman) A homological generalized Property R conjecture is false. [arXiv]

2. (with Román Aranda) Manifolds with weakly reducible genus-three trisections are standard. [arXiv]

Publications:

1. (with Filip Misev and Stefan Friedl) Bounding the ribbon numbers of knots and links, to appear in Alg. & Geom. Top. [arXiv]

2. (with B. Clingenpeel, Z. Dai, G. Diraviam, K. Jaber, K. Kar, Z. Liu, T. Miklethun, H. Nagampoozhy, M. Perry, M. Samuelson-Lynn, E. Seamans, A. Wright, N. Xie, and R. Zou) Colorings of symmetric unions and partial knots,  to appear in J Knot. Theory Ramifications. [arXiv]

3. (with Jeffrey Meier) Knots bounding non-isotopic ribbon disks, J Topol. 18 (2025), no. 4, Paper No. e70047, 18 pp. [arXiv] [journal]

4. Hexagonal lattice diagrams for complex curves in CP^2, Trans. Amer. Math. Soc. 378 (2025), no. 4, 2271-2319. [arXiv] [journal]

5. (with X. An, M. Aronin, D. Cates, A. Goh, B. Kirn, J. Krienke, M. Liang, S. Lowery, E. Malkoc, J. Meier, M. Natonson, V. Radic, Y. Rodoplu, B. Saha, E. Scott, and R. Simkins), Ribbon numbers of 12-crossing knots, J. Knot Theory Ramifications 34 (2025), no. 13, Paper No. 2550066, 45 pp. [arXiv] [journal]

6. (with Jason Joseph, Jeffrey Meier, and Maggie Miller) Bridge trisections and Seifert solids,  Alg. & Geom. Top. 25 (2025), 1501-1522. [arXiv] [journal]

7. (with Ethan Romary) A family of Andrews-Curtis trivializations via 4-manifold trisections, Geom. Dedicata 218 (2024), no. 2, 15 pp. [arXiv] [journal]

8. (with Maggie Miller) Equivalent characterizations of handle-ribbon knots, Comm. Anal. Geom. 31 (2023), no. 9, 2157-2193. [arXiv] [journal]

9. (with Jeffrey Meier and Abigail Thompson)  Cubic graphs induced by bridge trisections, Math. Res. Lett. 30 (2023), no. 4, 1207-1231. [arXiv] [journal]

10. (with W. Allred, M. Aragón, Z. Dooley, A. Goldman, Y. Lei, I. Martinez, N. Meyer, D. Peters, S. Warrander, and A. Wright) Tri-plane diagrams for simple surfaces in S^4,  J. Knot Theory Ramifications 32 (2023), no. 6, Paper No. 2350041, 28 pp. [arXiv] [journal]

11. (with Jeffrey Meier) Generalized square knots and homotopy 4-spheres,  J. Differential Geom. 122 (2022), no. 1, 69-129 [arXiv] [journal]

12. (with Jason Joseph, Jeffrey Meier, and Maggie Miller) Bridge trisections and classical knotted surface theory, Pacific J. Math 319 (2022), no. 2, 343-369. [arXiv] [journal]

13.  (with Rebecca Sorsen) The Kauffman bracket expansion of a generalized crossing, Top. Proc. 58 (2021), 298-301. [journal]

14. The Powell conjecture and reducing sphere complexes, J. London Math. Soc. (2)  101 (2020), 328-348. [arXiv] [journal]

15. (with Jeffrey Meier) Bridge trisections of knotted surfaces in 4-manifolds, Proc. Nat. Acad. Sci. 115 (2018), no. 43, 10880-10886. [arXiv] [journal]

16. (with Jeffrey Meier) Characterizing Dehn surgeries on links via trisections, Proc. Nat. Acad. Sci. 115 (2018), no. 43, 10887-10893. [arXiv] [journal]

17. (with Jeffrey Meier) Genus-two trisections are standard, Geom. Topol. 21 (2017), no. 3, 1583-1630. [arXiv] [journal]

18. (with Jeffrey Meier) Bridge trisections of knotted surfaces in S^4, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7343-7386. [arXiv] [journal]

19. (with Jessica Purcell) Independence of volume and genus g bridge numbers, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1805-1818. [arXiv] [journal]

20. (with Derek Davies) Natural properties of the trunk of the knot, J. Knot Theory Ramifications 26 (2017), no. 12, 1750080, 9 pp. [arXiv] [journal]

21. (with Jeffrey Meier and Trent Schirmer) Classification of trisections and the generalized property R conjecture, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4983-4997. [arXiv] [journal]

22. (with B. Jang, A. Kronaeur, P. Luitel, D. Medici, and S. A. Taylor) New examples of Brunnian theta graphs, Involve 9 (2016), no. 5, 857-875. [arXiv] [journal]

23. (with Samuel Taylor) Products of Farey graphs are totally geodesic in the pants graph, J. Topol. Anal. 8 (2016), no. 2, 287-311. [arXiv] [journal]

24. Uniqueness of higher genus bridge surfaces for torus knots, Math Proc. Cambridge Philos. Soc. 159 (2015), no. 1, 79-88. [arXiv] [journal]

25. (with Ryan Blair) Knots with compressible thin levels, Alg. Geom. Top. 15 (2015), no. 3, 1691-1715. [arXiv] [journal]

26. (with R. Sean Bowman and Scott A. Taylor) Bridge spectra of twisted torus knots, Int. Math. Res. Notices (2015), no. 16, 7336-7356. [arXiv] [journal]

27. Bridge spectra of iterated torus knots, Comm. Anal. Geom. 22 (2014), no. 5, 931-963. [arXiv] [journal]

28. Infinite cardinalities in the Hausdorff metric geometry, Involve 7 (2014), 285-293. [journal]

29. Bridge and pants complexities of knots, J. London Math. Soc. (2) 87 (2013), 43-68. [arXiv] [journal]

30. A lower bound on the width of satellite knots, Top. Proc. 40 (2012), 179-188. [arXiv] [journal]

31. Unexpected local minima in the width complexes for knots, Alg. Geom. Top. 11 (2011), no. 2, 1097-1106. [arXiv] [journal]

32. Properties of knots preserved by cabling, Comm. Anal. Geom. 19 (2011), no. 3, 541-562. [arXiv] [journal]

33. (with C. Blackburn, K. Lund, S. Schlicker, and P. Sigmon) A missing prime configuration in the Hausdorff Metric Geometry, J. Geom. 92 (2009), no. 1-2, 28-59. [journal]