Marc Lackenby

Research interests: Topology, geometry & group theory

Editor of International Mathematical Research Notices
Editor of 'Groups, Geometry and Dynamics'
Former editor of 'Journal of the London Mathematical Society'
Former Managing Editor of 'Journal of Topology'

Mathematical Institute office: N3.01
Phone (Institute): 01865 273561
Phone (College): 01865 271728
Email: lackenby(at)maths.ox.ac.uk

My CV (June 2024)

• 1. The Whitney trick, Topology and Its Applications 71 (1996) 115-118.
• 2. Fox's congruence classes and the quantum-SU(2) invariants of links in 3-manifolds, Comment. Math. Helv. 71 (1996) 664-677.
• 3. Surfaces, surgery and unknotting operations, Math. Ann. 308 (1997) 615-632.
• 4. Dehn surgery on knots in 3-manifolds, J. Amer. Math. Soc. 10 (1997) 835-864.
• 5. Upper bounds in the theory of unknotting operations, Topology 37 (1998) 63-73.
• 6. (With Daryl Cooper) Dehn surgery and negatively curved 3-manifolds, J. Differential Geom. 50 (1998) 591-624. (PS) (PDF)
• 7. Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243-282. (PS) (PDF)
• 8. Taut ideal triangulations of 3-manifolds, Geom. Top. 4 (2000) 369-395. (PS) (PDF)
• 9. Attaching handlebodies to 3-manifolds, Geom. Top. 6 (2002) 889-904. (PS) (PDF)
• 10. Exceptional surgery curves in triangulated 3-manifolds, Pacific J. Math. 210 (2003), 101-163. (PS) (PDF)
• 11. The canonical decomposition of once-punctured torus bundles, Comment. Math. Helv. 78 (2003) 363-384. (PS) (PDF)
• 12. The volume of hyperbolic alternating link complements Proc. London Math. Soc. 88 (2004) 204-224 (PS) (PDF)
• 13. The asymptotic behaviour of Heegaard genus Math. Res. Lett. 11 (2004) 139-149 (PS) (PDF)
• 14. The Heegaard genus of amalgamated 3-manifolds Geom. Dedicata 109 (2004) 139-145 (PS) (PDF)
• 15. A characterisation of large finitely presented groups J. Algebra 287 (2005) 458-473 (PS) (PDF)
• 16. Expanders, rank and graphs of groups Israel J. Math. 146 (2005) 357-370 (PS) (PDF)
• 17. Classification of alternating knots with tunnel number one Comm. Anal. Geom. 13 (2005) 151-186 (PS) (PDF)
• 18. Heegaard splittings, the virtually Haken conjecture and Property (tau), Invent. Math. 164 (2006) 317-359 (PS) (PDF)
• 19. Covering spaces of 3-orbifolds Duke Math J. 136 (2007) 181-203 (PS) (PDF)
• 20. Some 3-manifolds and 3-orbifolds with large fundamental group Proc. Amer. Math. Soc. 135 (2007) 3393-3402 (PS) (PDF)
• 21. Adding high powered relations to large groups Math. Res. Lett. 14 (2007) 983-993 (PS) (PDF)
• 22. (With Darren Long and Alan Reid) Covering spaces of arithmetic 3-orbifolds Int. Math. Res. Not. (2008) (PS) (PDF)
• 23. An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary Alg. Geom. Top. 8 (2008) 911-934 (PS) (PDF)
• 24. (With Darren Long and Alan Reid) LERF and the Lubotzky-Sarnak conjecture Geom. Topol. 12 (2008) 2047-2056 (PS) (PDF)
• 25. New lower bounds on subgroup growth and homology growth Proc. London Math. Soc. 98 (2009) 271-297. (PS) (PDF)
• 26. Large groups, Property (tau) and the homology growth of subgroups Math. Proc. Camb. Phil. Soc. 146 (2009) 625-648 (PS) (PDF)
• 27. The crossing number of composite knots J. Topology 2 (2009) 747-768. (PS) (PDF)
• 28. Surface subgroups of Kleinian groups with torsion Invent. Math. 179 (2010) 175-190. (PS) (PDF)
• 29. (with Daryl Cooper and Jessica Purcell) The length of unknotting tunnels Alg. Geom. Top. 10 (2010) 637-661. (PDF)
• 30. Spectral geometry, link complements and surgery diagrams Geom. Dedicata 147 (2010) 191-206. (PS) (PDF)
• 31. Detecting large groups J. Algebra 324 (2010) 2636-2657. (PS) (PDF)
• 32. (with Alexander Coward) Unknotting genus one knots Comment. Math. Helv. 86 (2011) 383-399. (PS) (PDF)
• 33. Finite covering spaces of 3-manifolds Proceedings of the ICM, Volume II, 1042-1070, Hindustan Book Agency, New Delhi, 2010 (PDF)
• 34. (with Robert Meyerhoff) The maximal number of exceptional Dehn surgeries Invent. Math. 191 (2013) 341-382. (PS) (PDF)
            The main theorems of this paper have a computer-assisted proof. The relevant C++ routines are below:
               slopes1.cpp,    slopes2.cpp,    slopes3.cpp,    slopes4.cpp,    slopes5.cpp,    myDouble.h,    myDouble.cpp
  
• 35. Finding disjoint surfaces in 3-manifolds Geom. Dedicata 170 (2014) 385-401 (PDF)
• 36. Core curves of triangulated solid tori Trans. Amer. Math. Soc. 366 (2014) 6027-6050 (PDF)
• 37. The crossing number of satellite knots Alg. Geom. Top. 14 (2014) 499-529 (PDF)
• 38. (With Alexander Coward) An upper bound on Reidemeister moves Amer. J. Math. 136 (2014) 1023-1066 (PDF)
• 39. (with Jessica Purcell) Geodesics and compression bodies Experimental Math. 23 (2014) 218-240 (PS) (PDF)
• 40. A polynomial upper bound on Reidemeister moves Annals Math. 182 (2015) 491-564 (PDF)
• 41. (with Jessica Purcell) Cusp volumes of alternating knots Geom. Topol. 20 (2016) 2053-2078. (PDF)
• 42. (with Jessica Purcell) Essential twisted surfaces in alternating link complements Alg. Geom. Topol. 16 (2016) 3209-3270 (PDF)
• 43. Elementary knot theory Lectures on Geometry (Clay Lecture Notes), Oxford University Press (2017) (PDF)
• 44. Some conditionally hard problems on links and 3-manifolds Discrete Comput. Geom. 58 (2017) 580-595. (PDF)
• 45. (with Andras Juhasz) Appendix to Alternating links and definite surfaces (by Joshua Greene) Duke Math J. 166 (2017) 2133-2151 (PDF)
• 46. Every knot has characterising slopes Math. Ann. 374 (2019) 429-446. (PDF)
• 47. Algorithms in 3-manifold theory Surveys in Differential Geometry 25 (2020),163-213. (PDF)
• 48. The efficient certification of knottedness and Thurston norm, Adv. Math. 387 (2021) 107796. (PDF)
• 49. Links with splitting number one, Geom. Dedicata 214 (2021), 319-351. (PDF)
• 50. (with Alex Davies, Petar Velickovic, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomasev, Richard Tanburn, Peter Battaglia, Charles Blundell, Andras Juhasz, Geordie Williamson, Demis Hassabis and Pushmeet Kohli) Advancing mathematics by guiding human intuition with AI, Nature 600 (2021) 70-74.
• 51. (with Agnese Barbensi, Dorothy Buck and Heather Harrington) Double branched covers of knotoids Comm. Anal. Geom. 30 (2022) 1007-1057. (PDF)
• 52. (with Jessica Purcell) The triangulation complexity of fibred 3-manifolds, Geom. Topol. (accepted) (PDF)
• 53. (with Alex Davies, Andras Juhasz and Nenad Tomasev) The signature and cusp geometry of hyperbolic knots, Geom. Topol. (accepted) (PDF)
• 54. (with Andreea Deac and Petar Velickovic) Expander graph propagation, Proceedings of Machine Learning Research (Learning on Graphs 2022) (PDF)
• 55. (with Mehdi Yazdi) The computational complexity of determining knot genus in a fixed 3-manifold, Proc. London. Math. Soc. 126 (2023) 837-879. (PDF)
• 56. (with Patrick Callahan, Ilya Kapovich, Peter Shalen, and Robin Wilson) Wolfgang Haken, 1928-2022, Notices Amer. Math. Soc. 70 (2023) 1452-1467.
• 57. (with Jessica Purcell) The triangulation complexity of elliptic and sol 3-manifolds, Math. Ann. (accepted) (PDF)

• 58. Dehn surgery from a hyperbolic perspective, 29 pages (PDF)
• 59. (with Saul Schleimer) Recognising elliptic manifolds, 33 pages (PDF)
• 60. (with Mehdi Yazdi) Bounds for the number of moves between pants decompositions, and between triangulations, 43 pages (PDF)
• 61. Some fast algorithms for curves in surfaces, 44 pages (PDF)
• 62. (with Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, Andras Juhasz, Nenad Tomaev, Daniel Zheng) The unknotting number, hard unknot diagrams, and reinforcement learning, 29 pages (PDF)

Some documents by my students

• Homology in finite index subgroups, PhD thesis by Liam Wall, January 2010
• Constructing the Cayley graphs of F_d/D_n^p(F_d), Summer project by Henry Bradford, September 2010

Some of my talks

Topology seminar, Cambidge and Princeton, March 2022

• Knot theory and machine learning

Topology seminar, UC Davis, February 2021 and Oxford, March 2021

• Unknot recognition in quasi-polynomial time (slides) (handout)

Topology seminar, Jerusalem and Copenhagen, May 2021

• Unknot recognition in quasi-polynomial time (longer version, slides)

The Kirby-Scharlemann-Thompson fest, June 2018

• The triangulation complexity of 3-manifolds

Scottish Topology Seminar / Swiss Knots, Spring 2013

• A polynomial upper bound on Reidemeister moves

Clay Mathematics Conference, June 2012

• The Dehn surgery problem

LMS Poincare anniversary meeting, May 2012

• The Poincare conjecture

Talk at Fritz Grunewald's 60th birthday conference, April 2009

• Surface subgroups of arithmetic Kleinian groups

4 Talks at Gottingen, Autumn School on Geometric Group Theory, November 2008

• Surface subgroups in dimension 3, I
• Surface subgroups in dimension 3, II
• Surface subgroups in dimension 3, III
• Surface subgroups in dimension 3, IV

Talk at Thurston's 60th birthday conference, June 2007

• The crossing number of composite knots

3 Talks at Oxford, Hilary Term 2007

• The geometry and topology of finite index subgroups, I
• The geometry and topology of finite index subgroups, II
• The geometry and topology of finite index subgroups, III

3 Talks at the Institute of Advanced Study, Princeton
Workshop on Lie Groups, Representations and Discrete Mathematics,
14-18 November 2005


Property (tau) and Geometry, I, was given by Alex Lubotzky
• Property (tau) and Geometry, II
• Property (tau) and Geometry, III
• Property (tau) and Geometry, IV

3 Talks at the ICMS, Edinburgh, "3-Manifolds after Perelman", 13-17 March 2006

• Finite covers of 3-manifolds, I
• Finite covers of 3-manifolds, II
• Finite covers of 3-manifolds, III

Lecture Notes and Problem Sheets

Topology & Groups, Michaelmas 2008 (updated July 2018)

• Lecture notes
• Problem Sheet 1
• Problem Sheet 2
• Problem Sheet 3
• Problem Sheet 4
• Problem Sheet 5
• Problem Sheet 6
• Problem Sheet 7

Section c Algebraic Topology, Hilary 2001

• Question Sheet 1
• Question Sheet 2
• Question Sheet 3
• Question Sheet 4
• Question Sheet 5
• Question Sheet 6
• Question Sheet 7

Three-dimensional manifolds (Graduate Course, Michaelmas 1999)

• Chapters 1 and 2
• Chapter 3
• Chapters 4 and 5
• Chapters 6 and 7
• Chapters 8 to 12
• Examples Sheet 1
• Examples Sheet 2

Hyperbolic manifolds (Graduate Course, Hilary 2000)

• Introductory chapter on Riemannian manifolds (Very brief!) (PS) (PDF)
• Chapters 0 - 5 (PS) (PDF)
• Chapters 6 - 11 (PS) (PDF)
• Example Sheet 1 (from Part III course) (PS) (PDF)
• Example Sheet 2 (from Part III course) (PS) (PDF)

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