Stephan Tillmann

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco–Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.

Let M be a compact 3-manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3-manifolds obtained as even Dehn fillings of M. As an application, we characterise some infinite families of even Dehn fillings of M for which our method determines the complexity of their members up to an additive constant. The constant only depends on the size of a chosen triangulation of M , and the isotopy class of its boundary. We then show that, given a triangulation T of M with 2-triangle torus boundary, there exist infinite families of even Dehn fillings of M for which we can determine the complexity of the filled manifolds with a gap between upper and lower bounds of at most 13jT j C 7. This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of M , or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the 3-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure-eight knot, the pretzel knot P. 2; 3; 7/, and the trefoil.

A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3–manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3–manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.

The range of geometric configurations possible for an arbitrary coordination centre ABn, is a key aspect of stereoisomerism and molecular geometry in general. Despite this, much focus within Chemistry concerning ABn geometries has not taken a rigorous and holistic approach. Using a precise definition for configurations based upon the interior angles at the A-atom, we present a general method for describing ABn configuration spaces or their subspaces. The method uses angle-changing molecular vibrational normal modes as a basis to locally describe the spaces. A consequence of using a physically inspired method is that the configuration spaces map onto physical mechanisms (“polytopal rearrangements”). The space of local configurations and their subspaces are amenable to compactifications modelled on chemical processes. A contiguous symmetry based discretisation method that allows for the configuration spaces to be encoded as graphs is presented. Numerous worked examples are given.

A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3–manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these immersions. This is used to develop and implement an algorithm to compute the unit ball of the Thurston norm for cusped hyperbolic 3–manifolds of finite volume. As an application of independent interest, we give an upper bound on the minimal entropy of pseudo-Anosov maps of surfaces with number of cusps bounded linearly in genus.

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.