Brent Everitt
The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by Lorimer, Richardson and Rubinstein and Best.
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We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra.
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 octahedral case, we also identify which manifolds are complements of an augmented knotted trivalent graph and give the corresponding link. A (small version of) the Platonic census and the related improved algorithms have been incorporated into SnapPy. The census also comes in Regina forma…
Mixed-platonic 3-manifolds
本文研究了由多种正理想超球体构成的混合柏拉图3维流形，证明了无隐藏对称性的超曲面不存在。
We compute the Zariski closure of the Kontsevich-Zorich monodromy groups arising from certain square tiled surfaces that are geometrically motivated. Specifically we consider three surfaces that emerge as translation covers of platonic solids and quotients of infinite polyhedra, and show that the Zariski closure of the monodromy group arising from each surface is equal to a power of $\rm{SL}(2, \mathbb{R})$. We prove our results by finding generators for the monodromy groups, using a theorem of Matheus-Yoccoz-Zmiaikou that provides constraints on the Zariski closure of the groups (to obtain an…
We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of primitive lattice surfaces, we show that the Teichmüller curve of the unfolded dodecahedron has genus 131 with 19 cone singularities and 362 cusps. We provide both theoretical and rigorous computer-assisted proofs that there are no closed saddle connections on the surfaces associa…
The 3-strand braid group with torsion
本文通过轨道形形式体系为带挠率的三股辫群提供拓扑解释，揭示其与柏拉图立体之间的结构联系，弥补了Coxeter有限逐例检验证明的不足。
A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics $-1$ $\&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms …
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an~orthonormal complete basis fo…
Knotted solid tori in contact manifolds
该研究聚焦切触流形中的实心环面，探究纽结型宽度与最大Thurston-Bennequin不变量的关系并给出判定条件，还证明多类纽结型中存在大量此前仅在$S^3$中发现的不可加厚环面。
Let $M=W\cup_T V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a Seifert surface in $W$ is a meridian of the solid torus. This means that there is a degree-one map $f\colon M\to N$, pinching $W$ into a solid torus while fixing $V$. We prove that $g(M)\ge g(N)$, where $g(M)$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern kn…
We show that if a hyperbolic 3-manifold $M$ with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then $M$ is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifold contains an essential torus which meets the core of the attached solid torus minimally in at most two points. This completes the determination of best possible upper bounds for the distance between two exceptional Dehn fillings yielding essential small surfaces in all ten cases for large hyperbolic 3-manifolds.
Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic $C^\infty$ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the existence of such a non-singular extension.
The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-metric" and the hyperbolic metric. We make partial progress in the program, proving long-time existence, preservation of negative sectional curvature, curvature bounds, and integral convergence to hyperbolic for the metrics under consideration.
We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $ζ_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely b=9. We also study the rational invariant a/b for single-complex-place cusped manifolds, …
For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundar…
Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $Φ\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $Φ$ is an explicit Seifert hypersurface for $M$.
We give definitions of cohomology determinants for compact, connected, orientable 3-manifolds. We also give formulae relating cohomology determinants before and after gluing a solid torus along a torus boundary component. Cohomology determinants are related to Turaev torsion, though the author hopes that they have other uses as well.
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