This repository contains source code for finding counterexamples for one-vertex 3-manifold triangulations by using Pachner moves to remove distinguished edges.

The code is written to run using Regina's regina-python command-line program. For details on how to install Regina, visit https://regina-normal.github.io/. See below for usage notes.

We currently recognise three types of distinguished edges:

• Core edges: Deleting a solid torus neighbourhood of such an edge leaves behind another solid torus. The existence of a core edge implies that the triangulation represents a lens space.
• Tunnel edges: In an ideal triangulation of a knot complement, deleting a small regular neighbourhood of such an edge turns the knot complement into a genus-2 orientable handlebody. The existence of a tunnel edge implies that the knot is either trivial or has tunnel number one.
• Edges isotopic to Seifert fibres

The main scripts in this repository are:

• removeCore.py: Attempts to remove a core edge.
• removeTunnel.py: Attempts to remove a tunnel edge.
• removeFibre.py: Attempts to remove an edge isotopic to a Seifert fibre.
• simplifyNoCore.py: Attempts to simplify a triangulation without introducing new core edges.
• simplifyNoTunnel.py: Attempts to simplify a triangulation without introducing new tunnel edges.
• simplifyNoFibre.py: Attempts to simplify a triangulation without introducing new edges isotopic to Seifert fibres.

The bulk of the supporting code can be found in search.py, removeBadEdge.py and simplify.py.

The script removeCore.py takes three command-line arguments:

• An isomorphism signature for the triangulation from which we would like to remove a core edge.
• The number of concurrent processes to use.
• The number of seconds to wait before printing a brief progress update.

In the following example, assume that the regina-python command-line program is in the directory dir. To use 4 concurrent processes to remove a core edge from the triangulation with isomorphism signature "cMcabbgqs", and to request a progress update every 10 seconds, run the following:

dir/regina-python removeCore.py cMcabbgqs 4 10

The scripts removeTunnel.py and removeFibre.py are similar, except that removeFibre.py takes two additional command-line arguments:

• The first additional argument is mandatory, and must be either ult or int. This option controls how the search measures the complexities of the triangulations that it encounters.

  • If the argument is ult, then the search tries to simultaneously minimise the complexity of all bad edges, since the ultimate goal is to remove all such edges.
  • If the argument is int, then the search instead focuses on an intermediate goal: reducing the number of bad edges by one.

  For details, see the complexity() routine in removeBadEdges.py.

• The second additional argument is optional. If provided, this must be a comma-separated list (without whitespace, e.g. 1,6,8) of integers corresponding to indices of bad edges. In this context, an edge is bad if it might be isotopic to a Seifert fibre; we emphasise that we do not conclusively test whether edges really are isotopic to Seifert fibres. This means that even if we have already identified edges in the initial triangulation that are not bad, we cannot reliably recompute which edges these are; the optional argument allows us to include this information as part of the input.

The scripts simplifyNoCore.py, simplifyNoTunnel.py and simplifyNoFibre.py each take four command-line arguments:

• An isomorphism signature for the triangulation that we would like to simplify.
• The number of concurrent processes to use.
• The number of seconds to wait before printing a brief progress update.
• The allowed excess height (i.e., the amount by which we are allowed to increase the size of the input triangulation).

— Alex He (a.he@uqconnect.edu.au)