Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognit…
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14 results for “Regina”
Algorithm finds knot friends in 3-manifolds.
problem Identifying knots that share similar 3-manifold complements.
method Developed an algorithm using SnapPy and Regina.
result Constructed a census of simple knots with friends.
Proves Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings.
problem Every link is 3-move equivalent to a trivial link.
method Computational methods, including new code in Regina.
result Proves conjecture for links with up to 20 crossings.
Algorithm constructs triangulations for Heegaard splittings and related 3-manifolds.
problem Constructing triangulations for Heegaard splittings and related 3-manifolds.
method Algorithm using Regina to generate triangulations from combinatorial presentations of Heegaard diagrams.
result Triangulations with cutwidth bounded by for genus- Heegaard splittings.
We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orienta…
Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an…
The paper generates triangulations of 2-knot complements via spinning 1-knots.
problem Generating triangulations of 2-knot complements.
method Algorithm to generate triangulations of 2-knot complements obtained by spinning 1-knots.
result Triangulations of exteriors of 2-knots from all 1-knots with up to eight crossings.
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it…
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…
Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.
problem Estimating the Heegaard genus of 3-manifolds.
method Turaev-Viro state sum TQFT and unitary modular category.
result Provides a lower bound for Heegaard genus using TQFT.
We developed an efficient algorithm to factorize knots.
problem Computing the prime factorization of knots efficiently.
method Introduced an edge-ideal triangulation to represent knots and developed an algorithm using Regina.
result Our algorithm works well for knots up to 19 crossings and provides new complexity results.
The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coord…
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
problem Differentiating and classifying 3-manifolds and their Dehn surgeries.
method Training machine learning models on isomorphism signatures derived from 3-manifold triangulations and Pachner graphs.
result Gradient saliency analysis reveals key parts of the language-like encoding scheme.
Computer program finds FAMED triangulations for thousands of knots.
problem Proving the Andersen-Kashaev volume conjecture for many knots.
method Straightforward computer implementation in Regina and Snappy.
result Andersen-Kashaev conjecture proven for over 42,000 knots.