相关论文: Isogonal conjugation in isosceles tetrahedron
A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2\pi}{3}$ but less than $2\pi$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the…
In the present paper we study hyperbolic manifolds that are rational homology 3-spheres obtained by colouring of right--angled polytopes. We study the existence (or absence) of different kinds of symmetries of rational homology spheres that…
We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing…
We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the…
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic…
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds.…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
We construct topological triangulations for complements of $(-2,3,n)$-pretzel knots and links with $n\ge7$. Following a procedure outlined by Futer and Gu\'eritaud, we use a theorem of Casson and Rivin to prove the constructed…
A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group $\mathrm{PSL}(2, \mathbb{C})$ of…
For hyperbolic 3-manifolds, the growth rate of their Turaev-Viro invariants, evaluated at a certain root of unity, is conjectured to give the hyperbolic volume of the manifold. This has been verified for a handful of examples and several…
We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we…
In a recent paper Garoufalidis and Reid constructed pairs of 1-cusped hyperbolic 3-manifolds which are isospectral but not isometric. In this paper we extend this work to the multi-cusped setting by constructing isospectral but not…
We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the…
We prove that a pair of continuous disjoint periodic curves in $\mathbb{C}$ inscribes an isosceles trapezoid with any similarity type. The case of smooth curves can be identified with a Lagrangian intersection problem for a pair of…
We obtain a rigidity result for compact three-dimensional Heterotic solitons with parallel non-trivial torsion. We show that they are either hyperbolic three-manifolds or compact quotients of the Heisenberg group equipped with a…
We show how the exceptional isogenies of classical groups to orthogonal groups of quadratic spaces of dimensions up to 8 over fields of characteristic different from 2 may be obtained by explicit algebraic constructions using the…
It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite…
This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly,…
The first main results of this note establish forms of the hyperbolic laws of cosines and sines for certain classes of quadrilaterals and pentagons in the hyperbolic plane, having at least one ideal vertex and right angles at non-ideal…