Jessica Purcell: Research

My research is in a branch of geometry, called low dimensional topology.

Specifically, I work with 3-manifolds: objects that look like 3-dimensional space when you zoom in far enough. By powerful results proved by Grigori Perelman soon after the turn of the 21st century, every 3-manifold can be cut into pieces that admit a geometry, and the most common geometry is hyperbolic. I look at questions relating hyperbolic geometry to more topological or combinatorial descriptions of 3-manifolds. Here are some questions I have investigated:

• A knot gives a 3-manifold: the knot complement, by removing the knot from 3-space. Given only a diagram of a knot, what is the hyperbolic volume of the knot? How does it depend on the diagram?

• Where are the geodesics (the straight lines) in knot diagrams and other simple spaces? Are important arcs in a topological description also geodesic under the geometric description?

• How complicated can the geometry of a knot complement be? What about knots obtained by topological moves from other knot complements? How are their geometries related?

Related Videos:

• A short, general-audience intro to my research, from Monash University Faculty of Science, 2019: Meet Professor Jessica Purcell (2:47 minutes)

• A public lecture for all audiences: The Mathematics of Knots (59:30 minutes)
  The above was a public lecture for ACEMS: the ARC Centre of Excellence for Mathematical and Statistical Frontiers, 1 April 2021.
  
  

• A longer talk on knot theory and hyperbolic geometry for undergraduates, also likely suitable for high school students: Hyperbolic Geometry and Knots (48:45 minutes)
  I gave the above talk at a joint online seminar of the Monash LunchMaths group and Melbourne University Maths and Stats Society (MUMS), 2020.
  
  

• Three introductory talks on my research area, pitched at the level of postgraduate students or advanced undergraduates, given at CIRM 2018. Each lecture is roughly one hour long.

  • Structure of hyperbolic manifolds -- Lecture 1

  • Structure of hyperbolic manifolds -- Lecture 2

  • Structure of hyperbolic manifolds -- Lecture 3

1. Alternating links on nonorientable surfaces and Klein-bottly alternating links
   With Lecheng Su
   Preprint (15 pages):[pdf]. arXiv:2412.07133.
   
   

2. Triangulations of the 3-sphere with knotted edge
   With Dionne Ibarra, Daniel Mathews, and Jonathan Spreer
   Preprint (19 pages):[pdf]. arXiv:2411.18938.

3. Polynomial bounds for surfaces in cusped 3-manifolds
   With Anastasiia Tsvietkova
   Preprint (26 pages):[pdf]. arXiv:2311.08567.

4. Rigidity of highly twisted plat diagrams
   With Nir Lazarovich, Yoav Moriah, Tali Pinsky, Jessica S. Purcell
   Preprint (36 pages): arXiv:2508.07303
   
   

5. On Geometric triangulations of double twist knots
   With Dionne Ibarra and Daniel Mathews
   Preprint (27 pages): arXiv:2504.09901

1. Augmented links, shadow links, and the TV volume conjecture: a geometric perspective
   With Dionne Ibarra and Emma McQuire
   New York Journal of Mathematics, to appear
   Preprint (23 pages): arXiv:2506.09296

2. Excluding cosmetic surgeries on hyperbolic 3-manifolds
   With David Futer and Saul Schleimer
   Journal of Computational Geometry, to appear
   Preprint (29 pages):[pdf]. arXiv:2403.10448.

3. Standard position for surfaces in link complements in arbitrary 3-manifolds
   With Anastasiia Tsvietkova
   Algebraic and Geometric Topology, to appear
   Preprint (39 pages): [pdf]. arXiv:2205.06368.

4. Hyperbolic and satellite Lorenz links obtained by twisting
   With Thiago de Paiva
   Michigan Mathematical Journal, to appear
   Preprint version (21 pages): [pdf]. arXiv:2301.01934.

5. Volume bounds for hyperbolic rod complements in the 3-torus
   With Norman Do and Connie On Yu Hui
   Pacific Journal of Mathematics, Vol. 339 (2025), No. 1, 167–189
   Preprint (19 pages):[pdf]. arXiv:2409.02357.

6. A symplectic basis for 3-manifold triangulations
   With Daniel V. Mathews
   Communications in Analysis and Geometry, Vol. 33 (2025), No. 4, pp. 815-878
   Preprint version (45 pages): [pdf]. arXiv:2208.06969.

7. A-polynomials, Ptolemy equations and Dehn filling
   With Josh Howie and Dan Mathews
   Algebraic and Geometric Topology, Algebraic & Geometric Topology 25 (2025) 1265–1320
   Preprint version (46 pages): [pdf]. ArXiv.

8. Geometry of fully augmented links in doubled 3-manifolds
   With Corbin Reid and John Stewart

Journal of Knot Theory and its Ramifications, Vol. 34, No. 06, 2550017 (2025)
Preprint (15 page):[pdf]. arXiv:2310.18976.

9. Large volume fibred knots of fixed genus
   With Kenneth L. Baker, David Futer and Saul Schleimer
   Mathematical Research Letters, Vol. 31 (2024), no. 5, 1305-1314.
   Preprint version (6 pages): [pdf]. Arxiv.

10. Alternating links on surfaces and volume bounds
    With Effie Kalfagianni
    Communications in Analysis and Geometry, Vol. 32 (2024), no. 1, 119-151.
    Preprint version (24 pages): [pdf]. ArXiv.

11. The triangulation complexity of elliptic and sol 3-manifolds
    With Marc Lackenby
    Mathematische Annalen, Vol. 290 (2024), no. 2, 1623-1667.
    Preprint version (41 pages): [pdf]. Arxiv.

12. The triangulation complexity of fibred 3-manifolds
    With Marc Lackenby
    Geometry & Topology, Vol. 28 (2024), no. 4, 1727-1828.
    Preprint (77 pages): [pdf]. ArXiv.

13. On the geometry of rod packings in the 3-torus
    With Connie On Yu Hui
    Bulletin of the London Mathematical Society, Vol. 56 (2024), No. 4, 1291-1309
    Preprint version (20 pages): [pdf]. Arxiv.

14. Arithmetic modular links
    With Tali Pinsky and Jose Andres Rodriguez-Migueles
    Pacific Journal of Mathematics, Vol. 327 (2023), No. 2, 337-358.
    Preprint version (18 pages): [pdf]. ArXiv.

15. A-Polynomials of fillings of the Whitehead sister
    With Joshua Howie, Daniel Mathews, and Em Thompson
    International Journal of Mathematics, Vol. 34 (2023), No. 13, 2350085.
    Preprint version (18 pages):[pdf]. ArXiv.

16. Satellites and Lorenz knots
    With Thiago de Paiva
    International Mathematics Research Notices (IMRN), Vol. 2023 (2023), no. 19, 16540-16573.
    Preprint version (24 pages): [pdf]. ArXiv.

17. Constructing knots with specified geometric limits
    With Urs Fuchs, John Stewart
    Pacific Journal of Mathematics, Vol. 324 (2023), No. 1, 111-142.
    Preprint version (27 pages): [pdf]. ArXiv.

18. Geometric triangulations and highly twisted links
    With Sophie Ham
    Algebraic and Geometric Topology, Vol. 23 (2023), 1399-1462.
    Preprint version (40 pages): [pdf]. ArXiv.

19. Effective drilling and filling of tame hyperbolic 3-manifolds
    With David Futer and Saul Schleimer
    Commentarii Mathematici Helvetici, Vol. 97 (2022), No. 3, 457-512.
    Preprint version (36 pages):[pdf]. ArXiv.

20. Effective bilipschitz bounds on drilling and filling
    With David Futer and Saul Schleimer
    Geometry & Topology, Volume 26 (2022), No. 3, 1077-1188.
    Preprint version (72 pages): [pdf]. ArXiv.

21. Right-angled polyhedra and alternating knots
    With Abhijit Champanerkar and Ilya Kofman
    Algebraic & Geometric Topology, Volume 22 (2022), No. 2, pp 739-784.
    Preprint version (32 pages): [pdf]. ArXiv.

22. State graphs and fibered state surfaces
    With Darlan Girao
    Algebraic and Geometric Topology, Volume 20 (2020), No. 2, pp 987-1014.
    Preprint version (20 pages): [pdf]. ArXiv.

23. Geometry of alternating links on surfaces
    With Josh Howie
    Transactions of the American Mathematical Society, Volume 373 (2020), No. 4, pp 2349-2397.
    Preprint version (44 pages): [pdf]. ArXiv.

24. Treewidth, crushing, and hyperbolic volume
    With Clement Maria
    Algebraic and Geometric Topology, Volume 19 (2019), No. 5, pp 2625-2652.
    Preprint version (20 pages): [pdf]. ArXiv.

25. Effective distance between nested Margulis tubes
    With David Futer and Saul Schleimer
    Transactions of the American Mathematical Society, Volume 372 (2019), No. 6, pp 4211-4237.
    Preprint version (25 pages): [pdf]. ArXiv.

26. On the tree-width of knot diagrams
    With Arnaud de Mesmay, Saul Schleimer, and Eric Sedgwick
    Journal of Computational Geometry, Volume 10 (2019), No. 1, pp 164-180.
    Preprint (14 pages): [pdf]. ArXiv.

27. A survey of hyperbolic knot theory (Book chapter)
    With David Futer and Efstratia Kalfagianni
    Knots, Low-Dimensional Topology and Applications, Springer Proceedings in Mathematics & Statistics, Volume 284 (2019), pp 1-30.
    Preprint version (20 pages): [pdf]. ArXiv.

28. Geometry of biperiodic alternating links
    With Abhijit Champanerkar and Ilya Kofman
    Journal of the London Mathematical Society, Vol. 99 (2019), No. 3, pp. 807-830.
    Preprint version (25 pages): [pdf]. ArXiv.

29. Cusp shape and tunnel number
    With Vinh Dang
    Proceedings of the American Mathematical Society, Vol. 147 (2019), pp 1351-1366.
    Preprint version (15 pages): [pdf]. ArXiv.

30. Spacious knots
    With Autumn Kent
    Mathematical Research Letters, Vol. 25 (2018), No. 2, pp 581-595.
    Preprint version (10 pages): [pdf]. ArXiv.

31. Geometry of planar surfaces and exceptional fillings
    With Neil Hoffman.
    Bulletin of the London Mathematical Society, Vol. 49 (2017), No. 2, pp. 185-201.
    Preprint version (17 pages): [pdf]. ArXiv.

32. Independence of volume and genus g bridge numbers
    With Alex Zupan
    Proceedings of the American Mathematical Society, Vol. 145 (2017), No. 4, 1805-1818.
    Preprint version (14 pages): [pdf]. ArXiv.

33. Geometrically and diagrammatically maximal knots
    With Abhijit Champanerkar and Ilya Kofman.
    Journal of the London Mathematical Society, Vol. 94 (2016), No. 3, 883-908.
    Preprint version (26 pages): [pdf]. ArXiv.

34. Volume bounds for weaving knots
    With Abhijit Champanerkar and Ilya Kofman.
    Algebraic and Geoemtric Topology, Vol. 16 (2016), No. 6, 3301-3323.
    Preprint version (17 pages): [pdf]. ArXiv.

35. Essential twisted surfaces in alternating link complements
    With Marc Lackenby.
    Algebraic and Geometric Topology, Vol. 16 (2016), No. 6, 3209-3270.
    Preprint version (51 pages): [pdf]. ArXiv.

36. Cusp volumes of alternating knots
    With Marc Lackenby.
    Geometry and Topology, Vol. 20 (2016), No. 4, 2053-2078.
    Preprint version (21 pages): [pdf]. ArXiv.

37. Bi-twist manifolds and two-bridge knots
    With J. W. Cannon, W. J. Floyd, L. Lambert, W. R. Parry
    Pacific Journal of Mathematics, Vol. 284 (2016), No. 1, 1-39.
    Preprint version (28 pages): [pdf]. ArXiv.

38. Density spectra for knots
    With Abhijit Champanerkar and Ilya Kofman.
    Journal of Knot Theory and its Ramifications, Vol. 25 (2016), no. 3, 164001, 11 pages.
    Preprint (8 pages): [pdf]. ArXiv.

39. Volumes of Montesinos links
    With Kathleen Finlinson.
    Pacific Journal of Mathematics. Vol. 282 (2016), no. 1, 63-105.
    Preprint version (41 pages): [pdf]. ArXiv.

40. Hyperbolic semi-adequate links
    With David Futer and Efstratia Kalfagianni.
    Communications in Analysis and Geometry. Vol. 23 (2015), no. 5, 993-1030.
    Preprint version (25 pages): [pdf]. ArXiv.

41. Volume and geometry of homogeneously adequate knots
    With Paige Bartholomew, Shane McQuarrie, and Kai Weser.
    Journal of Knot Theory and its Ramifications, Vol. 24 (2015), no. 8, 1550044, 29 pages.
    Preprint version (26 pages):[pdf]. ArXiv.

42. Geodesics and compression bodies
    With Marc Lackenby.
    Experimental Mathematics, Vol. 23 (2014), no. 2, 218-240.
    Preprint (31 pages): [pdf]. ArXiv.

43. Geodesic systems of tunnels in hyperbolic 3-manifolds
    With Stephan D. Burton.
    Algebraic and Geometric Topology, Vol. 14 (2014), no. 2, 925-952.
    Preprint version (19 pages): [pdf]. ArXiv.

44. Quasifuchsian state surfaces
    With David Futer and Efstratia Kalfagianni.
    Transactions of the American Mathematical Society, Vol. 366 (2014), no. 8, 4323-4343.
    Preprint version (21 pages): [pdf]. ArXiv.

45. Jones polynomials, volume, and essential knot surfaces: a survey
    With David Futer and Efstratia Kalfagianni.
    Proceedings of Knots in Poland III, Banach Center Publications, Vol. 100 (2014), pages 51-77.
    Preprint version (27 pages): [pdf]. ArXiv.

46. Dehn filling and the geometry of unknotting tunnels
    With Daryl Cooper and David Futer.
    Geometry and Topology, Vol. 17 (2013), no. 3, 1815-1876.
    Preprint version (45 pages): [pdf]. ArXiv.

47. Explicit Dehn filling and Heegaard splittings
    With David Futer.
    Communications in Analysis and Geometry, Vol. 21 (2013), no. 3, 625-650.
    Preprint version (17 pages): [pdf]. ArXiv.

48. Guts of surfaces and the colored Jones polynomial
    With David Futer and Efstratia Kalfagianni.
    Lecture Notes in Mathematics, Vol. 2069, Springer, Berlin, 2013.
    Preprint version (179 pages): [pdf]. ArXiv.

49. Volumes of chain links
    With James Kaiser and Clint Rollins.
    Journal of Knot Theory and Its Ramifications, Vol. 21 (2012), No. 11, 1250115, 17 pp.
    Preprint version (14 pages): [pdf]. ArXiv.
    Computational code and data available here.

50. Volume bounds for generalized twisted torus links
    With Abhijit Champanerkar, David Futer, Ilya Kofman, and Walter Neumann.
    Mathematical Research Letters, Vol. 18 (2011), no. 6, 1097--1120.
    Preprint version (21 pages): [pdf]. ArXiv.

51. An introduction to fully augmented links,
    Interactions between hyperbolic geometry, quantum topology, and number theory, Contemporary Mathematics, Vol 541, (2011), pp. 205--220.
    Preprint version (16 pages): [pdf].

52. Slopes and colored Jones polynomials of adequate knots
    With D. Futer and E. Kalfagianni.
    Proceedings of the American Mathematical Society, Vol. 139 (2011), no. 5, pp. 1889-1896.
    Preprint version (7 pages): [pdf]. ArXiv.

53. Geometric limits of knot complements
    With Juan Souto.
    Journal of Topology, Vol 3 (2010), no. 4, pp. 759-785.
    Preprint version (37 pages): [pdf]. ArXiv.

54. Cusp areas of Farey manifolds and applications to knot theory
    With D. Futer and E. Kalfagianni.
    International Mathematics Research Notices, Vol 2010 (2010), no. 23, pp. 4434-4497.
    Preprint version (44 pages): [pdf]. ArXiv.

55. On multiply twisted knots that are Seifert fibered or toroidal
    Communications in Analysis and Geometry. Vol 18 (2010), no. 2, pp. 219-256.
    Preprint version (32 pages): [pdf]. ArXiv.

56. On diagrammatic bounds of knot volumes and spectral invariants
    With David Futer and Efstratia Kalfagianni.
    Geometriae Dedicata, Vol. 147 (2010), no. 1, pp. 115-130.
    Preprint version (16 pages): [pdf]. ArXiv.

57. Hyperbolic geometry of multiply twisted knots
    Communications in Analysis and Geometry. Vol 18 (2010), no. 1, pp. 101-120.
    Preprint version (17 pages): [pdf]. ArXiv.

58. The length of unknotting tunnels
    With Daryl Cooper and Marc Lackenby.
    Algebraic & Geometric Topology, Vol. 10 (2010) pp. 637-661.
    Preprint version (20 pages): [pdf]. ArXiv.

59. Symmetric links and Conway sums: volume and Jones polynomial.
    With D. Futer and E. Kalfagianni.
    Mathematical Research Letters, Vol. 16 (2009), no. 2, pp. 233-253.
    Preprint version (18 pages): [pdf], ArXiv.

60. Cusp shapes under cone deformation.
    Journal of Differential Geometry, Vol. 80 (2008), no. 3, pp. 453-500.
    Preprint version (46 pages): [pdf], ArXiv.

61. Slope lengths and generalized augmented links.
    Communications in Analysis and Geometry, Vol. 16 (2008), no. 4, pp. 883-905.
    Preprint version (20 pages): [pdf], ArXiv.

62. Dehn filling, volume, and the Jones polynomial.
    With D. Futer and E. Kalfagianni.
    Journal of Differential Geometry, Vol. 78 (2008), no. 3, pp. 429-464.
    Preprint version (34 pages): [pdf], ArXiv.

63. Links with no exceptional surgeries.
    With David Futer.
    Commentarii Mathematici Helvetici, Vol. 8 (2007), no. 3, pp. 629-664.
    Preprint version (29 pages): [pdf], ArXiv.

64. Volumes of highly twisted knots and links.
    Algebraic & Geometric Topology, Vol. 7 (2007), pp. 93-108.
    Preprint version (14 pages): [pdf], ArXiv.

Publications In Computer Science (as Jessica Shepherd)

1. Riloff, E. and Shepherd, J., (1999) "A Corpus-Based Bootstrapping Algorithm for Semi-Automated Semantic Lexicon Construction", Journal of Natural Language Engineering, 1999, Vol. 5, No. 2, pp. 147-156.

2. Riloff, E. and Shepherd, J. (1997) "A Corpus-Based Approach for Building Semantic Lexicons", Proceedings of the Second Conference on Empirical Methods in Natural Language Processing (EMNLP-2).

I have written a book: An introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. It has three parts. The first part covers basic tools in hyperbolic geometry and geometric structures on 3-manifolds. The second part focuses on families of knots and links that have been amenable to study via hyperbolic geometry, particularly twist knots, 2-bridge knots, and alternating knots. It also develops geometric techniques used to study these families, such as angle structures and normal surfaces. The third part gives more detail on three important knot invariants that come directly from hyperbolic geometry, namely volume, canonical polyhedra, and the A-polynomial.

Comments, corrections, feedback welcome.

Update: The book is now available from the AMS Bookstore:
Hyperbolic Knot Theory

Or a draft version is available on the arXiv. The arxiv version has a few known errors that should be fixed in the published version.