Benjamin Burton — Code

Last updated: 1 October 2013

This page contains supporting software and data for a variety of papers:

Computational topology (various papers)
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
Computing closed essential surfaces in knot complements
Computing the crosscap number of a knot using integer programming and normal surfaces
Locating regions in a sequence under density constraints
The Weber-Seifert dodecahedral space is non-Haken

Computational topology (various papers)

Almost all of my computational topology code is included in Regina, a software package for 3-manifold topologists.

Regina: Software for 3-manifold topology and normal surface theory

You can download Regina from regina.sourceforge.net.

Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

This paper gives algorithms for enumerating fundamental normal surfaces in a 3-manifold triangulation, and uses these to compute previously-unknown crosscap numbers of knots. You can download a CSV file containing the new crosscap numbers here:

Download fund_results.csv (last updated 1 October 2013)

You should be able to open this file in your favourite spreadsheet application. The columns in this table are:

The crosscap numbers computed in this paper and the paper below (“Computing the crosscap number of a knot using integer programming and normal surfaces”) cover different knots, and so the two sets of downloads provide different information.

All KnotInfo data were retrieved on 30 September 2013.

Computing closed essential surfaces in knot complements

This paper (joint with Alexander Coward and Stephan Tillmann) includes a significant amount of supporting data for computations on the 2977 non-trivial prime knots with ≤12 crossings, as well as the two 20-crossing dodecahedral knots.

You can download two files with supporting data:

You can open each of these files in Regina to examine the ideal triangulations of the knot complements, as well as the lists of admissible extreme rays of the corresponding cones Q0(T). See the paper for details on how these data feature in the tests for whether these knots are large.

For convenience, you can also download the list of all 1019 large knots from the census of non-trivial prime knots with ≤12 crossings (i.e., the list of knots from the table at the end of the paper). This is a plain text file, and the knots within it are listed using their names from the KnotInfo database.

Finally, you can download the source code that produced this data:

Most, but not all, of this code is thoroughly documented, and the zip file will be updated again when the documentation is complete. In the meantime, see the comments in the code for further information.

Computing the crosscap number of a knot using integer programming and normal surfaces

This paper (joint with Melih Ozlen) gives three algorithms for computing the crosscap number of a knot and/or reducing the number of possible solutions, and uses the final algorithm to improve data in existing knot tables. You can download a CSV file containing this new data here:

You should be able to open this file in your favourite spreadsheet application. Most of the new data are for non-alternating knots, so you may need to scroll down a little way before you see these new results. The columns in this table are:

The crosscap numbers computed in this paper and the paper above (“Enumerating fundamental normal surfaces: Algorithms, experiments and invariants”) cover different knots, and so the two sets of downloads provide different information.

All KnotInfo data were retrieved on 2 June 2011.

You can also download a Regina data file containing full normal coordinates for each spanning surface found by these algorithms. These normal surfaces are, in effect, certificates that allow third parties to verify that the new crosscap numbers and upper bounds reported here are correct.

Download crosscap_data.rga

This data file matches version 2 of the CSV file above, and can be opened using Regina (a software package for 3-manifold topology with rich support for the enumeration and analysis of normal surfaces).

Locating regions in a sequence under density constraints

This paper (joint with Mathias Hiron) describes a number of sequence processing algorithms. You can download C++ implementations of these algorithms here:

Download density.zip

The programs in this archive search through long strings of 0s and 1s for regions of particular interest. They include:

The archive also includes a file README.txt with instructions for building and using these programs.

All of these programs are offered under the GNU General Public License.

Some of these programs use van Emde Boas trees, the implementation of which is taken from the MIT-licensed libveb by Jani Lahtinen. This is available from code.google.com, but the necessary portions are also included in the archive above.

The Weber-Seifert dodecahedral space is non-Haken

This paper (joint with J. Hyam Rubinstein and Stephan Tillmann) includes a significant amount of supporting data, including the 23-tetrahedron triangulation of the Weber-Seifert dodecahedral space and its 1751 standard vertex normal surfaces.

You can download this data from the Regina website.

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