# Professor Benjamin Burton

## Personal page

Professor Benjamin Burton's personal page

## Teaching and learning

Professor Burton teaches 1st year discrete mathematics, and 3rd year coding and cryptography, and set theory.

## Researcher biography

Benjamin Burton's research interests include computational geometry and topology, combinatorics, and information security. He also maintains an active role in gifted-and-talented programmes for secondary school students.

Benjamin Burton's research involves a blend of techniques from pure mathematics and computer science. His main interest is in computational geometry and topology in three and four dimensions, looking at problems such as how a computer can recognise whether a loop of string is knotted, or how it can identify large-scale geometric structures in a three-dimensional space. He is the primary author of the open source software package Regina, which implements state-of-the-art algorithms in this field.

His multi-disciplinary background includes a PhD in geometry and topology, an honours degree in combinatorics, research experience in information security, and three years as a research analyst in the finance industry. He has worked at several universities in Australia and overseas.

He maintains a strong interest in enrichment programmes for gifted and talented high school students, including the Mathematics and Informatics Olympiads and the National Mathematics Summer School. From 1999 until 2008 he directed the Australian training programme for the International Olympiad in Informatics (IOI), and from 2009 to 2014 he holds a seat on the international IOI Scientific Committee.

Benjamin is an active member of the UQ Ally Network, an award-winning program that supports and celebrates diversity of sexuality, gender and sex at UQ and in the broader community.

Featured projects | Duration |
---|---|

Projects in combinatorial geometry | |

Projects in computational topology |

### Book Chapter

*Computational topology with regina: algorithms, heuristics and implementations*. Geometry and topology down under: a conference in honour of Hyam Rubinstein. (pp. 195-224) edited by Craig D. Hodgson, William H. Jaco, Martin G. Scharlemann and Stephan Tillmann. Providence, RI, United States: American Mathematical Society. doi: 10.1090/conm/597/11877

### Journal Articles

*Arc diagrams on 3-manifold spines*. Discrete and Computational Geometry, 71 (4), 1190-1209. doi: 10.1007/s00454-023-00539-4

*Hard Diagrams of the Unknot*. Experimental Mathematics, 1-19. doi: 10.1080/10586458.2022.2161676

*Flip graphs of stacked and flag triangulations of the 2-sphere*. Electronic Journal of Combinatorics, 29 (2) P2.6. doi: 10.37236/10292

*Foreword*. Olympiads in Informatics, 16, 1-2. doi: 10.15388/ioi.2022.00

*On the hardness of finding normal surfaces*. Journal of Applied and Computational Topology, 5 (4), 583-619. doi: 10.1007/s41468-021-00076-0

*Embeddings of 3-Manifolds in S4 from the Point of View of the 11-Tetrahedron Census*. Experimental Mathematics, 31 (3), 1-26. doi: 10.1080/10586458.2020.1740836

*The parameterized complexity of finding a 2-sphere in a simplicial complex*. SIAM Journal on Discrete Mathematics, 33 (4), 2092-2110. doi: 10.1137/18M1168704

*Algorithms and complexity for Turaev–Viro invariants*. Journal of Applied and Computational Topology, 2 (1-2), 33-53. doi: 10.1007/s41468-018-0016-2

*Courcelle's theorem for triangulations*. Journal of Combinatorial Theory. Series A, 146, 264-294. doi: 10.1016/j.jcta.2016.10.001

*A construction principle for tight and minimal triangulations of manifolds*. Experimental Mathematics, 27 (1), 22-36. doi: 10.1080/10586458.2016.1212747

*Combinatorial Seifert fibred spaces with transitive cyclic automorphism group*. Israel Journal of Mathematics, 214 (2), 741-784. doi: 10.1007/s11856-016-1330-9

*On the complexity of immersed normal surfaces*. Geometry and Topology, 20 (2), 1061-1083. doi: 10.2140/gt.2016.20.1061

*Parameterized complexity of discrete Morse theory*. ACM Transactions On Mathematical Software, 42 (1) 2738034, 6:1-6:24. doi: 10.1145/2738034

*2-manifold recognition is in logspace*. Journal of Computational Geometry, 7 (1), 70-85.

*Separation index of graphs and stacked 2-spheres*. Journal of Combinatorial Theory. Series A, 136, 184-197. doi: 10.1016/j.jcta.2015.07.001

*A new approach to crushing 3-manifold triangulations*. Discrete and Computational Geometry, 52 (1), 116-139. doi: 10.1007/s00454-014-9572-y

*A duplicate pair in the SnapPea census*. Experimental Mathematics, 23 (2), 170-173. doi: 10.1080/10586458.2014.886535

*Multi-objective integer programming: an improved recursive algorithm*. Journal of Optimization Theory and Applications, 160 (2), 470-482. doi: 10.1007/s10957-013-0364-y

*Optimising a nonlinear utility function in multi-objective integer programming*. Journal of Global Optimization, 56 (1), 93-102. doi: 10.1007/s10898-012-9921-4

*Locating regions in a sequence under density constraints*. SIAM Journal on Computing, 42 (3), 1201-1215. doi: 10.1137/110830605

*Computing the crosscap number of a knot using integer programming and normal surfaces*. ACM Transactions On Mathematical Software, 39 (1) 4, 4.1-4.18. doi: 10.1145/2382585.2382589

*The Weber-Seifert dodecahedral space is non-Haken*. Transactions of the American Mathematical Society, 364 (2), 911-932. doi: 10.1090/S0002-9947-2011-05419-X

*Triangulating a Cappell-Shaneson knot complement*. Mathematical Research Letters, 19 (5), 1117-1126. doi: 10.4310/MRL.2012.v19.n5.a12

*Pachner moves, generic complexity, and randomising 3-manifold triangulations*. Oberwolfach Reports, 9 (2), 1412-1414. doi: 10.4171/OWR/2012/24

*Searching a bitstream in linear time for the longest substring of any given density*. Algorithmica, 61 (3), 555-579. doi: 10.1007/s00453-010-9424-y

*Maximal admissible faces and asymptotic bounds for the normal surface solution space*. Journal of Combinatorial Theory: Series A, 118 (4), 1410-1435. doi: 10.1016/j.jcta.2010.12.011

*Optimizing the double description method for normal surface enumeration*. Mathematics of Computation, 79 (269), 453-484. doi: 10.1090/S0025-5718-09-02282-0

*Quadrilateral-octagon coordinates for almost normal surfaces*. Experimental Mathematics, 19 (3), 285-315. doi: 10.1080/10586458.2010.10390625

*Get involved! The IOI workshop 2010, its goals and results*. Olympiads in Informatics, 4, 158-169.

*A mathematician reflecting on the International Olympiad in Informatics*. Australian Mathematical Society Gazette, 37 (1), 15-21.

*Converting between quadrilateral and standard solution sets in normal surface theory*. Algebraic and Geometric Topology, 9 (4), 2121-2174. doi: 10.2140/agt.2009.9.2121

*Breaking the routine: Events to complement informatics olympiad training*. Olympiads in Informatics, 2, 5-15.

*Creating informatics olympiad tasks: Exploring the black art*. Olympiads in Informatics, 2, 16-36.

*Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find*. Discrete and Computational Geometry, 38 (3), 527-571. doi: 10.1007/s00454-007-1307-x

*Informatics olympiads: Approaching mathematics through code*. Mathematics Competitions, 20 (2), 29-51.

*Structures of small closed non-orientable 3-manifold triangulations*. Journal of Knot Theory and Its Ramifications, 16 (5), 545-574. doi: 10.1142/S0218216507005439

*Observations from the 8-tetrahedron nonorientable census*. Experimental Mathematics, 16 (2), 129-144. doi: 10.1080/10586458.2007.10128994

*Efficient enumeration of 3-manifold triangulations*. The Australian Mathematical Society Gazette, 31 (2), 111-117.

*Introducing Regina, the 3-manifold topology software*. Experimental Mathematics, 13 (3), 267-272.

*Face pairing graphs and 3-manifold enumeration*. Journal of Knot Theory and Its Ramifications (JKTR), 13 (8), 1057-1101. doi: 10.1142/S0218216504003627

### Conference Papers

*Finding large counterexamples by selectively exploring the Pachner graph*.

*Symposium on Computational Geometry (SoCG)*, Dallas, TX, United States, 12-15 June 2023. Wadern, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. doi: 10.4230/LIPIcs.SoCG.2023.21

*The next 350 million knots*.

*36th International Symposium on Computational Geometry (SoCG 2020)*, Zürich, Switzerland, 23-26 June 2020. Wadern, Germany: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH. doi: 10.4230/LIPIcs.SoCG.2020.25

*Knot Diagrams of Treewidth Two*.

*46th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2020)*, Online, 24–26 June 2020. Heidelberg, Germany: Springer. doi: 10.1007/978-3-030-60440-0_7

*The HOMFLY-PT polynomial is fixed-parameter tractable*.

*34th International Symposium on Computational Geometry, SoCG 2018*, Budapest, Hungary, 11-14 June 2018. Wadern, Germany: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. doi: 10.4230/LIPIcs.SoCG.2018.18

*The parameterized complexity of finding a 2-sphere in a simplicial complex*.

*34th Symposium on Theoretical Aspects of Computer Science, STACS 2017*, Hannover, Germany, 8 - 11 March 2017. Wadern, Germany: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH. doi: 10.4230/LIPIcs.STACS.2017.18

*Randomness testing and comparison of classical and quantum bit generators*.

*2017 IEEE Symposium on Computers and Communications (ISCC),*, Heraklion, Greece, 3-6 July 2017. Piscataway, NJ United States: Institute of Electrical and Electronics Engineers . doi: 10.1109/ISCC.2017.8024660

*Computing optimal homotopies over a spiked plane with polygonal boundary*.

*25th European Symposium on Algorithms, ESA 2017*, Vienna, Austria, 4-6 September 2017. Wadern, Germany: Schloss Dagstuhl - Leibniz-Zentrum fur Informatik GmbH. doi: 10.4230/LIPIcs.ESA.2017.23

*Finding Non-orientable Surfaces in 3-Manifolds*.

*32nd Annual ACM International Symposium on Computational Geometry (SoCG)*, Boston Ma, Jun 14-17, 2016. New York, NY United States: Springer New York. doi: 10.1007/s00454-017-9900-0

*Finding non-orientable surfaces in 3-manifolds*.

*International Symposium on Computational Geometry*, Boston, MA, United States, 14-17 June 2016. Wadern, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. doi: 10.4230/LIPIcs.SoCG.2016.24

*Efficient algorithms to decide tightness*.

*International Symposium on Computational Geometry*, Boston, MA, United States, 14-18 June 2016. Wadern, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. doi: 10.4230/LIPIcs.SoCG.2016.12

*Tabulation of 3-manifolds of lengths up to 10*.

*International Conference on Topology and Geometry 2013, joint with the Sixth Japan-Mexico Topology Symposium*, Matsue, Japan, September 2-6, 2013. Amsterdam, Netherlands: Elsevier BV. doi: 10.1016/j.topol.2015.05.036

*An edge-based framework for enumerating 3-manifold triangulations*.

*International Symposium on Computational Geometry*, Eindhoven, Netherlands, 22-25 June 2015. Wadern, Germany: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. doi: 10.4230/LIPIcs.SOCG.2015.270

*Algorithms and complexity for Turaev-Viro invariants*.

*International Colloquium on Automata, Languages, and Programming*, Kyoto, Japan, 6-10 June 2015. Heidelberg, Germany: Springer. doi: 10.1007/978-3-662-47672-7_23

*Fixed parameter tractable algorithms in combinatorial topology*.

*20th International Computing and Combinatorics Conference, COCOON 2014*, Atlanta, GA United States, 2 - 6 August 2014. Heidelberg, Germany: Springer. doi: 10.1007/978-3-319-08783-2_26

*Courcelle's theorem for triangulations*.

*Workshop on Triangulations in Geometry and Topology. SoCG 2014: Computational Geometry Week 2014. The 30th Annual Symposium on Computational Geometry*, Kyoto, Japan, 8-11 June, 2014. Ithaca, NY, USA: Cornell University Library.

*Enumerating fundamental normal surfaces: algorithms, experiments and invariants*.

*16th Workshop on Algorithm Engineering and Experiments (ALENEX14)*, Portland, United States, 5 January 2014. Philadelphia, United States: Society for Industrial and Applied Mathematics (SIAM). doi: 10.1137/1.9781611973198.11

*A new approach to crushing 3-manifold triangulations*.

*SoCG '13: Symposium on Computational Geometry 2013*, Rio de Janeiro, Brazil, 17 - 20 June 2013. New York, NY United States: Association for Computing Machinery. doi: 10.1145/2493132.2462409

*A tree traversal algorithm for decision problems in knot theory and 3-manifold topology*. Secaucus, NJ, United States: Springer. doi: 10.1007/s00453-012-9645-3

*Computing closed essential surfaces in knot complements*.

*29th Annual Symposium on Computational Geometry (SoCG 2013)*, Rio de Janeiro, Brazil, 17-20 June 2013. New York, NY United States: Association for Computing Machinery Inc.. doi: 10.1145/2462356.2462380

*Parameterized complexity of discrete morse theory*.

*29th Annual Symposium on Computational Geometry (SoCG 2013)*, Rio de Janeiro, Brazil, 17-20 June 2013. New York, NY, United States: Association for Computing Machinery Inc.. doi: 10.1145/2462356.2462391

*Computationally proving triangulated 4-manifolds to be diffeomorphic*.

*29th ACM Symposium on Computational Geometry,*, Rio de Janeiro, Brazil, 17 - 20 June 2013.

*The complexity of detecting taut angle structures on triangulations*.

*24th ACM-SIAM Symposium on Discrete Algorithms (SODA 2013)*, New Orleans, United States, 6-9 January 2013. Philadelphia, United States: Society for Industrial and Applied Mathematics. doi: 10.1137/1.9781611973105.13

*Computing closed essential surfaces in knot complements*.

*SoCG '13: Symposium on Computational Geometry 2013*, Rio de Janeiro, Brazil, 17 - 20 June 2013. New York, NY United States: Association for Computing Machinery. doi: 10.1145/2493132.2462380

*A new approach to crushing 3-manifold triangulations*.

*29th Annual Symposium on Computational Geometry (SoCG 2013)*, Rio de Janeiro, Brazil, 17-20 June 2013. New York, NY United States: Association for Computing Machinery Inc.. doi: 10.1145/2462356.2462409

*Complementary vertices and adjacency testing in polytopes*.

*18th Annual International Computing and Combinatorics Conference (COCOON 2012)*, Sydney, Australia, 20-22 August 2012. Heidelberg, Germany: Springer. doi: 10.1007/978-3-642-32241-9_43

*Computational topology and normal surfaces: theoretical and experimental complexity bounds*.

*Meeting on Algorithm Engineering and Experiments (ALENEX13)*, New Orleans, United States, 7 January 2013. Philadelphia, United States: Society for Industrial and Applied Mathematics. doi: 10.1137/1.9781611972931.7

*The Pachner graph and the simplification of 3-sphere triangulations*.

*27th ACM Symposium on Computational Geometry [SoCG]*, Paris, France, 13-15 June 2011. New York, NY, U.S.A.: ACM. doi: 10.1145/1998196.1998220

*A tree traversal algorithm for decision problems in knot theory and 3-manifold topology*.

*27th Annual Symposium on Computational Geometry (SoCG 2011)*, Paris, France, 13-15 June 2011. New York, NY, United States: ACM Press. doi: 10.1145/1998196.1998219

*Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations*.

*36th International Symposium on Symbolic and Algebraic Computation [ISSAC]*, San Jose, CA, United States, 8-11 June 2011. New York, NY, United States: ACM Press. doi: 10.1145/1993886.1993901

*Encouraging algorithmic thinking without a computer*. Lithuania: Institute of Mathematics and Informatics.

*The complexity of the normal surface solution space*.

*26th ACM Symposium on Computational Geometry [SCG]*, Snowbird, Utah, U.S.A., 13-16 June 2010. New York , U.S.A.: ACM (Association for Computing Machinery) Press. doi: 10.1145/1810959.1810995

*Informatics Olympiads: Challenges in programming and algorithm design*.

*Thirty-First Australasian Computer Science Conference (ACSC 2008)*, Wollongong, NSW, Australia, 22-25 Jan 2008. Sydney, Australia: Australian Computer Society (ACS).

*Secure group communication with distributed generation of private keys for ad-hoc networks*.

*IFIP TC11 20th IFIP International Information Security Conference*, Chiba, Japan, 3 May- 1 Jun 2005. New York, USA: Springer. doi: 10.1007/0-387-25660-1_31

### Data Collection

*Tabulation of knots and 3-manifolds*. The University of Queensland. (Dataset) doi: 10.48610/3bb4c69