Computational topology with Regina: Algorithms, heuristics and implementations

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 recognition and connected sum decomposition. This paper brings 3-manifold topologists up-to-date with Regina as it appears today, and documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance. These include the all-important simplification heuristics, key choices of data structures and algorithms to alleviate bottlenecks in normal surface enumeration, modern implementations of 3-sphere recognition and connected sum decomposition, and more. We also give some historical background for the project, including the key role played by Rubinstein in its genesis 15 years ago, and discuss current directions for future development.Comment: 29 pages, 10 figures; v2: minor revisions. To appear in "Geometry & Topology Down Under", Contemporary Mathematics, AM

Towards human readability of automated unknottedness proofs

© 2018 CEUR-WS. All rights reserved. When is a knot actually unknotted? How does one convince a human reader of the correctness of an answer to this question for a given knot diagram? For knots with a small number of crossings, humans can be efficient in spotting a sequence of untangling moves. However, for knot diagrams with hundreds of crossings, computer assistance is necessary. There have been recent developments in algorithms for both (indirectly) (i) detecting unknotedness and (directly) (ii) producing such sequences of untangling moves. Automated reasoning can be applied to (i) and, to some extent, (ii), but the computer output is not necessarily human-readable. We report on work in progress towards bridging the gap between the computer output and human readability, via generating human-readable visual proofs of unknottedness

Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants

Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold. In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds. The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants

Bending rigidity, supercoiling and knotting of ring polymers: models and simulations

The first part of the thesis was focussed on the interplay between knotting propensity and bending rigidity of equilibrated rings polymers. We found a surprising result: the equilibrium incidence of knots has a strongly non- monotonic dependence on bending, with a maximum at intermediate flexural rigidities. We next provided a quantitative framework, based on the balance of bending energy and configurational entropy, that allowed for rationalizing this counter-intuitive effect. We next extended the investigation to rings of much larger number of beads, via an heuristic model mapping between our semiflexible rings of beads and self-avoiding rings of cylinder. By the mapping, we not only confirmed the unimodal knotting profile for chains of 1,000 beads, but further found that chains of > 20,000 beads are expected to feature a bi-modal profile. We believe it would be most interesting to direct future efforts to confirm this transition from uni- to bi-modality using advanced sampling techniques for very long polymer rings. The second part of the thesis focused on the interplay of DNA knots and su- percoiling which are typically simultaneously present in vivo. We first studied this interplay by using oxDNA, an accurate mesoscopic DNA model and using it to study ings of thousands of base pairs tied in complex knots and with or without negative supercoiling (as appropriate for bacterial plasmids). By monitoring the dynamics of the DNA rings we found that the simultaneous presence of knots and supercoiling, and only their simultaneous presence, leads to a dramatic slowing down of the system reconfiguration dynamics. In particular, the essential tangles in the knotted region acquire a very long-lived character that, we speculate, could aid their recognition and simplification by topoisomerase. Finally, motivated by the recent experimental breakthrough that detected knots in eukaryotic DNA, we investigated the relationship between the compactness, writhe and knotting probability. The model was tuned to capture some of the salient properties of yeast minichromosomes, which were shown experimentally to become transiently highly knotted during transcription

Finding large counterexamples by selectively exploring the Pachner graph

We often rely on censuses of triangulations to guide our intuition in $3$-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to $10$ tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain $3$-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the $3$-manifold.Comment: 35 pages, 28 figures. A short version has been accepted for SoCG 2023; this full version contains some new results that do not appear in the SoCG versio

Triangulations

The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

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

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been variants of the classical double description method. In this paper we present the first practical normal surface enumeration algorithm that breaks out of the double description paradigm. This new algorithm is based on a tree traversal with feasibility and domination tests, and it enjoys a number of advantages over the double description method: incremental output, significantly lower time and space complexity, and a natural suitability for parallelisation. Experimental comparisons of running times are included

Discrete Differential Geometry

Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session

Elucidating the early events of protein aggregation using biophysical techniques

Proteins and peptides can convert from their native form into insoluble highly ordered fibrillar aggregates, known as amyloid fibrils. The process of fibrillogenesis is implicated in the pathogenic mechanisms of many diseases and, although mature fibrils are well characterised by a plethora of biophysical techniques, the initiation and early steps remain, to date, ambiguous. Mass spectrometry can provide invaluable insights into these early events as it can identify the low populated and transient oligomeric species present in the lag phase by their mass to charge ratio. Recent evidence has shown that oligomers formed early in the aggregation process are cytotoxic and may additionally be central to the progression of diseases associated with amyloid fibril presence. The hybrid technique of ion mobility mass spectrometry can be employed to provide conformational details of monomeric and multimeric species present and elucidate the presence of oligomers which possess coincident mass to charge ratios. Molecular modelling, in conjunction with experimental results, can suggest probable monomeric and oligomeric structural arrangements. In this thesis three aggregating systems are investigated: amyloidogenic transthyretin fragment (105-115), insulin and two Aβ peptides. Initially amyloidogenic endecapeptide transthyretin (105-115) is studied as it has been widely utilised as a model system for investigating amyloid formation due to its small size. Secondly insulin, a key hormone in metabolic processes, is investigated as extensive research has been carried out into its aggregation into amyloid fibrils. The formation of insulin amyloid fibrils rarely occurs in vivo; however localised amyloidosis at the site of injection and the aggregation of pharmaceutical insulin stocks present problems. Thirdly the aggregation of A β peptides Aβ (1-40) and Aβ (1-42) and their interactions with an aggregation inhibitor, RI-OR2, are characterised. A (1-42), although less commonly produced in vivo, is more cytotoxic and has a faster aggregation mechanism than Aβ (1-40). Both Aβ peptides are implicated in the aetiology of Alzheimer’s disease whilst RI-OR2 has been reported to prevent the production of high molecular weight oligomers, with particular suppression of Aβ (1-42) aggregation