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Small algorithm to compute sphere counts in Malcev coordinates. See for instructions on how to run.
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sphere words [a, b]

Heisenberg Group Sphere Count

Status: Polished.

This algorithms generate the sphere counts for the three-dimensional Heisenberg group over the integers (nilpotent of step 2), and gives the sphere count in terms of its Malcev coordinates.


To run these algorithms on your computer, you will need a running version of SageMath.

By following the commands below, you will obtain a sphere words [generating set] file (like the one shown in the repo), with the generating set of your choice.

To load, type in at the sage prompt in command line:

sage: load("spherecount_enumerate_coord_v1.sage")

To obtain the sphere words [generating set] file, for example, the sphere counts from radius 1 up to 8 with standard generating set [a,b] type in

sage: print_spherecounts([a,b], 8)

once the script is done running, the file shoud appear in the folder you're in from the command line!


This code was written as a research project at Tufts University under the supervision of Moon Duchin and Ayla P. Sánchez. I would also like to thank Charles Freeman for chatting with me, as well as tEp, Random and David Yang for giving me a place to crash during a couple blissfully dizzying months in Cambridge, MA.

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