Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
Method calculates polytope volume using graph combinatorics and Kirillov-Reshetikhin invariants.
problem Computing the volume of hyperbolic polyhedra.
method Combining combinatorial graph reductions and geometric splitting into tetrahedra.
result Volume of polytope can be expressed through critical values of a potential function.
Survey of methods for computing volumes of moduli spaces.
problem Computing volumes of moduli spaces for Riemann surfaces with different metrics.
method Combinatorial enumeration, intersection theory, recursion relations.
result Review of key results and methods in computing both Weil-Petersson and Masur-Veech volumes.
The paper proves combinatorial versions of systolic inequalities for manifolds.
problem Establishing inequalities for combinatorial structures of manifolds.
method Using triangulations and Riemannian metrics, the paper establishes combinatorial systolic inequalities.
result A class of manifolds satisfies a systolic inequality if and only if it satisfies a combinatorial systolic inequality.
Study proves hyperbolic structures for link complements in Seifert fibered spaces.
problem Proving hyperbolic structures for link complements in Seifert fibered spaces.
method Combinatorial bounds on volume of hyperbolic structures.
result Complement of a link in a Seifert fibered space admits a hyperbolic structure of finite volume.
Completed volumes match with combinatorial classes of the double ramification cycle.
problem Computing Masur-Veech volumes for quadratic differentials.
method Describing components of the double ramification cycle and their excess intersection classes, leading to a recursion for completed volumes.
result Completed volumes agree with top intersection of tautological classes on the double ramification cycle.
Study on random alternating link diagrams and their hyperbolic volumes.
problem Understanding the relationship between the combinatorial structure and hyperbolic volume of random links.
method Model based on random 4-valent maps, analyzing alternating and nonalternating diagrams.
result Expected hyperbolic volume is asymptotically linear in the number of crossings for random alternating diagrams.
Study shows infinite dimensional zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
problem Understanding the zero norm subspace in bounded cohomology of acylindrically hyperbolic groups.
method Introduced combinatorial volume forms and a new seminorm on exact bounded cohomology to construct non-trivial classes.
result Shows infinite dimensional zero norm subspace in degree 3 bounded cohomology of acylindrically hyperbolic groups.
Study measures volume of foliations on surfaces, finding integrability range.
problem Volume of combinatorial unit ball of measured foliations on bordered surfaces.
method Analyzes combinatorial moduli spaces and Kontsevich measure.
result Determines range of integrability for (BΣmcomb)s. We give a explicit computation of the pointed harmonic volumes of hyperelliptic curves with Weierstrass base points, which are paraphrased into a combinatorial formula.
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…
The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6-dimensions, are given, each of very small volume, …
This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.
problem Understanding the geometry of combinatorial Teichmüller space.
method Developed a parallel between combinatorial Teichmüller space and Weil-Petersson geometry, using measured foliations and Fenchel-Nielsen coordinates.
result Established a geometric recursion and topological recursion for mapping class group invariants.
Proves a conjecture about the maximum tet-volume of triangulations of a 2-sphere.
problem Proving the conjectured maximum tet-volume for all triangulations of a 2-sphere.
method Simplified version of Mathieu and Thurston's combinatorial proof using more general volume notions.
result Proves the full conjecture about the maximum tet-volume for all triangulations of a 2-sphere.
We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…
Proof confirms volume conjecture for a specific knot.
problem Verifying the volume conjecture for a specific knot.
method Using a generalized topological quantum field theory and a tetrahedral decomposition.
result Volume conjecture holds for the 73 knot in S3. We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxim…
This paper shows a relationship between 3-manifold complexity and systolic volume.
problem Understanding the complexity of 3-manifolds.
method Using systolic volume and triangulation complexity, the paper establishes a relationship between these two concepts.
result The systolic volume of a 3-manifold is related to its complexity.
We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds wit…
We find the asymptotic expansion of Masur-Veech volumes for large genus.
problem The asymptotic behavior of Masur-Veech volumes as genus increases.
method Combination of combinatorial and algebro-geometric approaches.
result Existence and computation of a complete asymptotic expansion.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both L2 combinatorial and L2 analytic torsion invariants …
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions…
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
Study floating bodies of polytopes, linking volume to flags.
problem Understanding floating bodies of polytopes in various spaces.
method Introducing flag simplices to connect metric and combinatorial structures.
result Weighted volume depends on complete flags of polytopes.
New formula calculates volumes and Chern-Simons invariants for closed 3-manifolds.
problem Computing volumes and Chern-Simons invariants for non-parabolic representations.
method Introducing deformed Ptolemy varieties to extend Zickert's formula.
result Volume and Chern-Simons invariants computed for closed 3-manifolds.
Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.
problem Calculating volumes of specific strata of quadratic differentials.
method Expressed volumes as a sum over stable graphs, with coefficients as intersection numbers of psi classes with combinatorial classes.
result Formula for volumes of odd strata of quadratic differentials.
Study of n-cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for n-cylinder contributions. Extends potential function to non-boundary parabolic representations for computing 3-manifold invariants.
problem Computing invariants of 3-manifolds from representations.
method Extends Cho and Murakami's potential function to non-boundary parabolic representations and derives combinatorial formulas.
result Combinatorial formulas for volume and Chern-Simons invariants of 3-manifolds.
The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.
problem Connecting surface volumes to polynomial evaluations of quantum invariants.
method Developing combinatorial and algebraic techniques to compute isomorphisms between representations.
result Numerical evidence supports a conjecture linking surface volumes to polynomial evaluations.
Software finds ideal polyhedra with rational dihedral angles and volume maxima.
problem Finding ideal convex polyhedra with maximal volume in hyperbolic 3-space.
method Rivin's variational characterization and combinatorial optimization algorithms.
result Maximal volume ideal polyhedra have dihedral angles that are rational multiples of π.
We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion c…
The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…
We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on CP1 and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
Study simplicial volume of manifolds from reflection group trick.
problem Characterize manifolds with positive simplicial volume.
method Define a partial order on triangulations and solve explicitly for minimal elements.
result Explicitly solved triangulations of the two-dimensional sphere and performed extensive analysis for three-dimensional case.
Smooth symplectic manifolds can be approximated by PL symplectic manifolds.
problem Understanding the relationship between smooth and piecewise linear symplectic structures.
method Defining PL symplectic manifolds and proving approximations.
result Smooth symplectic manifolds can be C0-approximated by PL symplectic manifolds. EDLP samples flat modes in discrete spaces using entropy.
problem Sampling flat modes in discrete spaces is challenging.
method EDLP uses a continuous auxiliary variable and local entropy to guide sampling.
result EDLP consistently outperforms traditional methods in various tasks.
New decompositions reveal volumes of tiling links in hyperbolic and spherical settings.
problem Understanding volumes of tiling links in different geometries.
method Generalizing angle structures and using truncated bipyramids to compute volumes.
result Volumes of hyperbolic tiling links are dense in a specific interval.
Paper studies metric ribbon graphs and provides a recursion for their volumes.
problem Calculating volumes of combinatorial moduli spaces of directed metric ribbon graphs.
method Decomposes directed ribbon graphs into simpler graphs with one vertex, proving a canonical recursion scheme for volumes.
result Explicit recursion for volumes of four-valent metric ribbon graphs provided.
The paper calculates large genus limits for quadratic differential volumes and constants.
problem Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials.
method Combining recursive relations (Virasoro constraints) and asymmetric simple random walk jump probabilities.
result Confirm predictions about Masur-Veech volumes and area Siegel-Veech constants.
New geometric decompositions reveal volumes of hyperbolic tiling links.
problem Understanding volumes of hyperbolic tiling links.
method Explicit geometric decompositions and bipyramid constructions.
result Volumes of hyperbolic tiling links are dense in a specific interval.
We calculate volumes of quadratic differentials using topological recursion.
problem Calculating volumes of quadratic differentials on curves.
method Topological recursion and geometric recursion applied to hyperbolic lengths of multicurves.
result Formula for constant terms of polynomials in terms of stable graphs.
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian man…