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It is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation, i.e. it is decomposed into positive volume ideal hyperbolic tetrahedra. In this talk, we prove that sufficiently highly twisted knots admit a geometric triangulation. In addition, by extending work of Gueritaud and Schleimer, we also give quantified versions of this result for infinite families of examples.
Given two one-vertex triangulations of the same 3-manifold, each with at least two tetrahedra, it is well known that these two triangulations must be connected by some finite sequence of 2-3 and 3-2 moves (the 2-3 move replaces two adjacent tetrahedra with three tetrahedra arranged around a common edge, and the 3-2 move is the inverse move). It would be nice to prove something stronger: that any two such triangulations are connected by a "monotonic" sequence of 2-3 and 3-2 moves. Here, "monotonic" means that the sequence is allowed to begin with a series of 2-3 moves, but only 3-2 moves are allowed after the first 3-2 move has occurred; in other words, once we decrease the number of tetrahedra for the first time, we are never allowed to increase this number again. In this talk, we discuss a partial result: there is always a sequence that is "semi-monotonic"; that is, the sequence is like a monotonic sequence, except we allow some 2-0 moves (which decrease the number of tetrahedra) to occur between the last 2-3 move and the first 3-2 move.
A geometric triangulation of a Riemannian manifold is a triangulation by totally geodesic simplexes. Any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves. In this talk I will present a bound on the length of this sequence, after taking a bounded number of barycentric subdivisions. These bounds are in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically triangulated closed hyperbolic or low dimensional spherical manifolds are isometric or not. This is joint work with Advait Phanse.
One could understand a great deal about a finite volume hyperbolic orbifold by studying properties of its number fields such as trace field, invariant trace field and cusp field. In this talk, I will talk about the (hyperbolic) orbifolds obtained by Dehn filling a fixed cusp of \(6^2_2\), a two component hyperbolic link. \(6^2_2\) is a tetrahedral link, i.e., its complement in \(S^3\) can be triangulated into regular ideal hyperbolic tetrahedra. I will illustrate how we can use this tetrahedral decomposition and some algebra to extract information about the number fields of the (hyperbolic) orbifolds resulting from Dehn filling a fixed cusp of \(6^2_2\). This talk will be based on joint work with Eric Chesebro, Jason DeBlois, Neil R Hoffman, Christian Millichap and William Worden.
The contractibility problem, in different types of spaces, is a fundamental problem in algorithmic topology. In this talk, I consider the complexity of deciding whether an arbitrary given closed curve on the boundary of a given 3-manifold is contractible in the 3-manifold. The same argument that shows the unknot recognition is in the class NP shows that this problem is in NP when the given curve is simple. I will present an overview of the proof that the problem lies in the class NP for an arbitrary given curve and 3-manifold. This is true, even if the given curve is represented in a compressed form.
I’ll discuss how symmetries inform the structure of the SL(2,C) character varieties of knot complements. Further, I’ll talk about how symmetries affect the detection of surfaces in the 3-manifold via the character variety. I’ll conclude with some examples. This is joint work with Jay Leach.
When one randomly glues a finite number of tetraheda together along their faces, the probability that the resulting complex is a manifold tends to zero as the number of tetrahedra grows. However, the only non-manifold points are the vertices of this complex. So, if we truncate the tetrahedra at their vertices, we obtain a random manifold with boundary. This talk will be about the geometry and topology of that manifold. This is joint work with Jean Raimbault.
The colored Jones polynomial is a generalization of the Jones polynomial from the finite-dimensional representations of \(U_q(sl_2)\). One motivating question in quantum topology is to understand how the polynomial relates to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the slopes of essential surfaces of a knot. As motivated by the recent progress on the conjecture, we discuss a connection from the colored Jones polynomial of a knot to the normal surface theory of its complement. We give a map relating generators of a state-sum expansion of the polynomial to normal subsets of a triangulation of the knot complement. Besides direct applications to the slope conjecture, we will also discuss applications to colored Khovanov homology.
We describe a new method of approximating a surface F in \(R^3\) by a high quality mesh, a piecewise-flat triangulated surface whose triangles are as close as possible to equilateral. The main advantage is improved mesh quality which is guaranteed for smooth surfaces. The GradNormal algorithm generates a triangular mesh that gives a piecewise-differentiable approximation of F, with angles between 35.2 and 101.5 degrees. As the mesh size approaches 0, the mesh converges to F through surfaces that are isotopic to F. This is joint work with J.Hass.
We will discuss necklace structures on cusped hyperbolic 3-manifolds arising as lifts of maximal cusp neighborhoods. Containing a short necklace forces many geometric and combinatorial restrictions on a manifold. In fact, we show that manifolds which admit necklace structures of length at most 7 are Dehn fillings of a known list of hyperbolic 3-manifolds. Combined with computer assisted results that prove the existence of short necklaces in manifolds with low cusp-volume, this allows us to answer questions about volumes of closed manifolds and exceptional Dehn fillings. At the end, we will discuss ongoing work to extend these results to distances between exceptional slopes.
By combining algorithms developed separately by Lackenby and Jaco, we provide an algorithm to construct 1-efficient triangulations for all but at most finitely many possible Dehn fillings of a specified triangulated knot manifold. This relies heavily notion of triangulated Dehn fillings developed by Jaco to have control over the normal surfaces added during the filling. After discussing some relevant background, I will describe how this method works and what triangulations it can be applied to.
I will discuss the classification of noncompact orientable surfaces and a new method to cut a general surface into simpler pieces. Then I will mention how this tool can be used to get general results about the mapping class groups of surfaces with noncompact boundary components.
We will discuss completeness properties of the \(L^p\) metrics on Teichmüller space. The Teichmüller space of a surface parametrizes the conformal structures it supports. By defining this space analytically we can equip it with the \(L^p\) metrics, of which the Teichmüller and Weil-Petersson metrics are special cases.
Results of Kordek and Margalit show the quotient of the braid group by the level 4 braid group is an extension of the symmetric group by \(Z^{\binom{n}{2}}_{2}\) under the standard action of \(S_n\) on unordered pairs. Eilenberg and MacLane proved extensions of a group can be classified by low dimensional cohomology of classifying spaces. In this talk I will begin by describing the connection between group cohomology and group extensions. Then I will constructing a representation for the 2-cocycle corresponding to this quotient of the braid group.