On the combinatorics of veering triangulations
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
In the 1970's and 1980's William Thurston revolutionised low-dimensional topology. He introduced many new ideas into the field, and furthermore revealed exciting connections between the combinatorial study of three-manifolds and many other areas:
Teichmuller theory, holomorphic dynamics, Kleinian groups, geometric group theory, and more.
One of the new techniques he pioneered was the use of ideal triangulations to represent hyperbolic three-manifolds. This allowed topologists, on the one hand, access to tools in algebraic geometry and, on the other hand, an organising principle for what had been a hodge-podge of disparate examples. The idea of ideal triangulations has now been generalised and specialised many times to give angled triangulations [Casson], taut ideal triangulations [Lackenby], and veering triangulations [Agol, HRST].
Francois Gueritaud has found an important connection between the Cannon-Thurston map for fibered manifolds and their veering triangulations. Yair Minsky and Samuel Taylor have given an interesting relation between the machinary of subsurface projections and the structure of veering triangulations. The goal of this project is to understand how the combinatorics of a veering triangulation, in the non-fibered setting, inform the cooresponding Cannon-Thurston map and the "fibered structure" of the universal cover of the three-manifold.
Teichmuller theory, holomorphic dynamics, Kleinian groups, geometric group theory, and more.
One of the new techniques he pioneered was the use of ideal triangulations to represent hyperbolic three-manifolds. This allowed topologists, on the one hand, access to tools in algebraic geometry and, on the other hand, an organising principle for what had been a hodge-podge of disparate examples. The idea of ideal triangulations has now been generalised and specialised many times to give angled triangulations [Casson], taut ideal triangulations [Lackenby], and veering triangulations [Agol, HRST].
Francois Gueritaud has found an important connection between the Cannon-Thurston map for fibered manifolds and their veering triangulations. Yair Minsky and Samuel Taylor have given an interesting relation between the machinary of subsurface projections and the structure of veering triangulations. The goal of this project is to understand how the combinatorics of a veering triangulation, in the non-fibered setting, inform the cooresponding Cannon-Thurston map and the "fibered structure" of the universal cover of the three-manifold.
Organisations
Publications
Parlak A
(2021)
Computation of the Taut, the Veering and the Teichmüller Polynomials
in Experimental Mathematics
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509796/1 | 01/10/2016 | 30/09/2021 | |||
| 1936817 | Studentship | EP/N509796/1 | 02/10/2017 | 16/04/2021 | Anna Barbara Parlak |
| Description | I have devised algorithms to compute two polynomial invariants of veering triangulations: the taut polynomial, and the veering polynomial (defined by Landry-Minsky-Taylor). In the special case of layered veering triangulations this gives an algorithm to compute the Teichmuller polynomial of a fibered face of the Thurston norm ball (defined by McMullen). The importance of this invariant follows from the fact that it encodes the stretch factors of monodromies of all fibrations lying over the same fibered face of the Thurston norm ball. There have been several other algorithms to compute this polynomial, but all of them worked only in some special cases, while the algorithm relying on veering triangulations is universal. Furthermore, it has been implemented and executed on thousands of examples, so researchers now have access to an abundant data set on Teichmuller polynomials. This will facilitate further research on the properties of stretch factors of pseudo-Anosov homeomorphisms. In fact, this data was already used by Tsang-Hironaka in their work on pseudo-Anosov maps with minimal stretch factor. I also proved that the taut polynomial of a veering triangulation is equal to the Alexander polynomial of the underlying manifold twisted by a certain representation of the fundamental group to Z/2. This result has surprisingly many consequences. 1) Algebraic properties of the taut polynomial can be deduced from algebraic properties of twisted Alexander polynomials. 2) The taut polynomial, like every twisted Alexander polynomial, can be computed from the presentation of the fundamental group of the manifold using Fox calculus. 3) For any 3-manifold M there are only finitely many potential candidates for the taut polynomial of a veering triangulation of M. This is an important observation because of an interesting conjecture that any 3-manifold admits only finitely many veering triangulations. If there is a 3-manifold with infinitely many veering triangulations, then infinitely many of them must have the same taut polynomial. |
| Exploitation Route | The implementation of the algorithm to compute the Teichmuller polynomials of fibered faces of the Thurston norm ball can be used by other researchers working on the dynamics of pseudo-Anosov homeomorphisms. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2101.12162 |
| Description | Oberwolfach Leibniz Graduate Students |
| Amount | € 200 (EUR) |
| Organisation | Mathematical Research Institute of Oberwolfach |
| Sector | Academic/University |
| Country | Germany |
| Start | 02/2020 |
| End | 02/2020 |
| Description | Research visit, host: Richard Kenyon |
| Amount | $100 (USD) |
| Organisation | Yale University |
| Sector | Academic/University |
| Country | United States |
| Start | 07/2019 |
| End | 07/2019 |
| Description | Research visit, host: Samuel Taylor |
| Amount | $83 (USD) |
| Organisation | Temple University |
| Sector | Academic/University |
| Country | United States |
| Start | 07/2019 |
| End | 07/2019 |
| Description | Workshop 'Perspectives on Dehn surgery' |
| Amount | £843 (GBP) |
| Organisation | Institute for Computational and Experimental Research in Mathematics |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 07/2019 |
| End | 07/2019 |