The paper studies triangulations of surfaces up to branched transit equivalences, proving equivalence conditions and foliations.
problem Understanding triangulations of surfaces up to branched transit equivalences.
method Analyzes triangulations of closed surfaces S with vertices V, considering branched triangulations up to b-transit equivalence generated by b-flips.
result Branched triangulations are equivalent under certain conditions, including parity of Euler-Poincare' characteristic c(S).
Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.
problem Equivalence of ideal triangulations in 3-manifolds.
method Using branched triangulations and transit equivalences, the paper shows that ideal triangulations are equivalent up to certain moves.
result Ideal triangulations of 3-manifolds are equivalent up to certain moves.
New method connects veering triangulations to dynamic pairs.
problem Understanding veering triangulations and their properties.
method Shearing decomposition of veering triangulations.
result Canonically associated dynamic pairs of branched surfaces.
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
Every open Riemann surface can be triangulated with equilateral triangles.
problem The structure and triangulation of Riemann surfaces.
method Constructing a holomorphic branched covering to the Riemann sphere and glueing together equilateral triangles.
result Every open Riemann surface can be equilaterally triangulated.
A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem …
Proofs for decomposing branched affine surfaces into triangles and cylinders.
problem Decomposing branched affine surfaces into simpler geometric shapes.
method Proof of Veech's theorem and introduction of invariant α.
result Any pair of decompositions can be connected by flips.
A universal branched 3-manifold W characterizes Sol 3-manifolds.
problem Characterizing Sol 3-manifolds
method Using a universal branched 3-manifold W and a regular language result A closed 3-manifold M immerses into W if and only if M admits a Sol structure. Study braid group actions on exceptional sequences using branched coverings.
problem Transitivity of braid group action on full exceptional sequences.
method Relate exceptional sequences to branched coverings, apply Birman--Hilden theory.
result Counterexamples to Bondal--Polishchuk conjecture on braid group transitivity.
Let M be a closed hyperbolic 3-manifold with a fibered face σ of the unit ball of the Thurston norm on H2(M). If M satisfies a certain condition related to Agol's veering triangulations, we construct a taut branched surface in M spanning σ. This partially answers a 1985 question of Oertel, and extends an e…
New method finds unique branched surfaces in 3-manifolds.
problem Finding unique branched surfaces in 3-manifolds.
method Proving endperiodic maps have periodic splitting sequences.
result Veering branched surfaces are unique up to equivalence.
Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.
problem Combinatorial descriptions of branched spines for 3-manifolds and their equivalence relations.
method Demonstrated that 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses.
result Simpler combinatorial descriptions for closed 3-manifolds and combed 3-manifolds.
LCD n-manifolds are linked to branched n-manifolds.
problem Characterizing compact n-manifolds locally.
method Equivalence between LCD and branched n-manifolds.
result LCD is equivalent to the existence of a branched n-manifold.
Solve arc diagrams on surfaces via branched covers.
problem Computing arc diagrams on surfaces via branched covers.
method Represent branched covers combinatorially and solve membership problem.
result Efficient solution for triangulated arc diagrams.
Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle Mf has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of 3--cell, we construct a different cell decomposition Df of Mf that is always realisable in …
The paper verifies hyperbolic structures on 3-manifolds using interval arithmetic.
problem Verifying hyperbolic structures on 3-manifolds using interval arithmetic.
method Extending methods by Casson, using interval arithmetic and a new theoretical result.
result Successfully verified hyperbolic structures on known examples and determined knots with hyperbolic branched covers.
Characterizes pseudo-Anosov orbit spaces via bifoliated planes
problem Characterizing actions on bifoliated planes arising from pseudo-Anosov flows
method Using branched covers, veering triangulations, and a compactness criterion
result Extends previous work on the special case with no odd-prong singularities
If all but two vertices of a triangulated sphere have degrees divisible by k, then the exceptional vertices are not adjacent. This theorem is proved for k=2 with the help of the coloring monodromy. For k=3,4,5 colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can asso…
We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate …
New train tracks for complex homeomorphisms found.
problem Existence of irreducible train tracks for pseudo-Anosov homeomorphisms.
method Starting from a veering triangulation, identify and modify branches to bypass obstructions.
result Construction of invariant train tracks with irreducible transition matrix.
The study connects triangulated surfaces to complex projective structures and circle patterns.
problem Understanding circle patterns on complex projective tori.
method Using discrete holomorphic quadratic differentials, the approach involves cross ratio systems and Delaunay angles.
result For any triangulated torus, the projection map is a covering map with at most one branch point.
Invariant of 3-manifolds using Hopf algebra elements.
problem Constructing an invariant for closed 3-manifolds.
method Using involutory Hopf algebra elements and normal o-graphs.
result Invariant values in cyclic quotient of Heisenberg double.
We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.
problem Existence of foliations transverse to pseudo-Anosov flows in 3-manifolds.
method Combinatorial approach via veering triangulations and branched surfaces.
result Existence of transverse foliations for certain pseudo-Anosov flows.
We organize the quantum hyperbolic invariants (QHI) of 3-manifolds into sequences of rational functions indexed by the odd integers N≥3 and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic 3-manifolds M we generalize the QHI and get rati…
We show that the correction terms in Heegaard Floer homology give a lower bound to the the genus of one-sided Heegaard splittings and the Z2--Thurston norm. Using a result of Jaco--Rubinstein--Tillmann, this gives a lower bound to the complexity of certain closed 3--manifolds. As an application, we compute…
A finite quiver Q without loops or 2-cycles defines a 3CY triangulated category D(Q) and a finite heart A(Q). We show that if Q satisfies some (strong) conditions then the space of stability conditions Stab(A(Q)) supported on this heart admits a natural family of semisimple Frobenius manifold structures, cons…
We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topologic…
We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can therefore be used to study the question of which foliations arise as geometric limits …
Geometrically interprets cup products and defines combinatorial Pin structures.
problem Understanding Steenrod's cup products and their geometric interpretation.
method Constructs vector fields and combinatorial frames to interpret cochain-level formulas.
result Geometrically interprets cup products and defines Pin structures combinatorially.
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…
Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
Minimal triangulations for 229 hyperbolic census knots discovered.
problem Finding minimal triangulations for hyperbolic census knots.
method Ideal triangulations of the magic manifold, low-complexity triangulations for partial fillings, sorting into families.
result Minimal triangulations for 229 hyperbolic census knots discovered, conjectured to be minimal for all 42 families.
Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, givin…
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
problem Finding the minimum number of tetrahedra in triangulations of 3-manifolds.
method Computed the triangulation complexity of all elliptic and sol 3-manifolds, within a bounded error.
result Computed the triangulation complexity of all elliptic and sol 3-manifolds.
With the [0,1,2]-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic q-equivelar triangulations of orientable and non-orientable surfaces for every q=3k, k≥2, and every q=3k+1, k≥3. Series of cy…