New triangulations for twist knots, proving volume conjecture.
problem Computing the volume of twist knot complements.
method Ideal triangulations and H-triangulations, using Thurston's method and volume functional.
result Proved the Teichmüller TQFT volume conjecture for all twist knots.
We utilize the obstruction theory of Galewski-Matumoto-Stern to derive equivalent formulations of the Triangulation Conjecture. For example, every closed topological manifold M^n with n > 4 can be simplicially triangulated if and only if the two distinct combinatorial triangulations of RP^5 are simplicially concordant.
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
Study proves volume conjecture for specific 3-manifolds.
problem Proving the Andersen-Kashaev volume conjecture for FAMED triangulations.
method Introducing FAMED triangulations and proving existence of Jones function.
result Proves the Andersen-Kashaev volume conjecture for FAMED geometric triangulations.
Proof outlines existence of non-triangulable manifolds in high dimensions.
problem Existence of non-triangulable manifolds in dimensions >4.
method Homology cobordism invariants from Pin(2) symmetry of Seiberg-Witten equations.
result Existence of non-triangulable manifolds proven in dimensions >4.
Proves ideal triangulations of hyperbolic alternating links are non-degenerate.
problem Proving ideal triangulations of hyperbolic alternating links are non-degenerate.
method Using non-positively curved cubings of prime alternating link exteriors.
result Ideal triangulations are non-degenerate, guaranteeing hyperbolicity equations have solutions.
The paper proves conditions for tight triangulations in 3-manifolds.
problem Characterizing tight triangulations in 3-manifolds.
method Analyzing properties of triangulations in terms of orientability, neighbourliness, and stacking.
result Triangulations of closed 3-manifolds are tight if they are orientable, neighbourly, and stacked.
Computer program finds FAMED triangulations for thousands of knots.
problem Proving the Andersen-Kashaev volume conjecture for many knots.
method Straightforward computer implementation in Regina and Snappy.
result Andersen-Kashaev conjecture proven for over 42,000 knots.
The paper confirms conjectures about the topology of triangulated polyhedra and geodesic triangulations on spheres.
problem Topology of spaces of convex polyhedra and Delaunay triangulations on spheres.
method Variational principles on triangulated surfaces.
result Spaces of Delaunay triangulations have the same homotopy types as their smooth counterparts on the unit 2-sphere.
Paper proves Luo's conjecture for 3D triangulated manifolds.
problem Finding hyperbolic metrics on compact 3-manifolds with boundary.
method Introduced and extended combinatorial Ricci flow to handle singularities.
result Proved Luo's conjecture affirmatively for ideal triangulations.
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.
The volume conjecture is proven for twist knots after Dehn filling.
problem Proving the volume conjecture for twist knots after Dehn filling.
method Constructing a new ideal triangulation of the Whitehead link complement.
result Chen-Yang's volume conjecture holds for sufficiently large parameters.
We find vertex bounds for triangulated manifolds and apply them to 4-manifold complexity.
problem Finding vertex bounds for triangulated manifolds in arbitrary dimensions.
method Analyzing face numbers and proving bounds for triangulations of manifolds.
result We prove tight bounds for odd-dimensional manifolds and conjecture for even dimensions, with applications to 4-manifold complexity.
Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.
problem Proving a conjecture about polynomial invariants and torsion for fibered 3-manifolds.
method Using combinatorial data of ideal triangulations and layered triangulations of fibered 3-manifolds with toroidal boundary, proving the conjecture for specific cases and confirming it for a large number of nonfibered manifolds.
result The conjecture linking 1-loop invariants to torsion for fibered 3-manifolds with toroidal boundary is proven.
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.
Authors find no positive spun triangulations for certain hyperbolic 3-manifolds.
problem Finding positive spun triangulations for hyperbolic 3-manifolds.
method Using Choi's result, they provide examples of closed hyperbolic 3-manifolds and geodesics without positive spun ideal triangulations.
result They provide evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.
New method finds large counterexamples by selectively exploring triangulations.
problem Finding small counterexamples in 3-manifold triangulations. method Selective enumeration of triangulations using heuristics.
result Found counterexamples to three conjectures about vertex triangulations.
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.
Highly twisted knots can be geometrically triangulated.
problem Proving geometric triangulations for hyperbolic 3-manifolds.
method Using highly twisted knots and extending Gueritaud and Schleimer's work.
result Sufficiently highly twisted knots admit a geometric triangulation.
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…
The paper examines how edge subdivisions affect the vanishing of L2-homology in Coxeter groups.
problem The vanishing of L2-homology in Coxeter groups under edge subdivisions. method Investigates conditions for the vanishing of L2-homology to be preserved under edge subdivisions of flag triangulations. result Conditions are given to preserve the vanishing of L2-homology under edge subdivisions, and counterexamples are constructed for a torsion growth analogue of Singer's conjecture. 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.
In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower bound conjecture for triangulated spheres in the context of the lower bound theorem…
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a ne…
Khovanov homology detects essential surfaces in knot complements.
problem Detecting essential surfaces in knot complements.
method Identifying Khovanov chain complex generators with normal surfaces using ideal triangulations.
result Colored Khovanov homology detects essential surfaces as in slope conjectures.
In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.
In the paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in [5]. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles…
The paper finds minimal contact triangulations for 3-manifolds, proving a linear bound on vertex count.
problem Finding the minimum number of vertices in contact triangulations for 3-manifolds.
method Explicit examples and linear growth bound analysis for overtwisted contact structures.
result The number of vertices for minimal contact triangulations grows at most linearly with respect to the d3 invariant. Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: 1) a minimal triangulation of a closed irreducible or a bounded hyperbolic 3-…
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
Classifies triangulated 4-manifolds up to six pentachora, finding exceptions.
problem Classifying all triangulated 4-manifolds up to a certain complexity.
method Framework to classify PL-types of triangulated 4-manifolds, using census approach.
result Findings exceptions for specific 4-manifolds, conjecturing they are standard.
This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex Σ_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of Σ_L is L and (2) Σ_L is contractible. It follows that if L is a triangulation of S^{n-1}, then Σ_L…
We create a minimal triangulation of 5D real projective space.
problem Tackling the minimal triangulation of 5D real projective space.
method Constructing a 6-dimensional polytope with a highly symmetric automorphism group.
result Our construction uses the fewest number of vertices (24) for a triangulation of 5D real projective space.
Lecture notes on monopole Floer homology, including differential geometry and correction terms.
problem Defining and understanding monopole Floer homology.
method Explains differential geometry, Morse theory, and four-dimensional theory connections.
result Sketches the relation to Manolescu's disproof of the Triangulation Conjecture.
We describe two methods for showing that a vector can not be the f-vector of a homology d-ball. As a consequence, we disprove a conjectured characterization of the f-vectors of balls of dimension five and higher due to Billera and Lee. We also provide a construction of triangulated balls with various f-vectors. We show…
We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurston's equation and Haken equation are made. Resolutions of these conjecture will lea…
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spinc structure isomorphic to its conjugate, we define the counterpart in this…
Triangulates permutahedra for Coxeter groups, revealing braid group connections.
problem Triangulating permutahedra for Coxeter groups.
method Constructs triangulations using total linear stability and height functions.
result Explicitly relates two braid group presentations.
The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.
problem Analyzing Teichmüller TQFT for hyperbolic knots with generalized FAMED triangulations.
method Introducing generalized FAMED property, proving exponential decay of partition functions in semi-classical limit.
result Partition functions decay exponentially with the volume of knot complements, and the 1-loop invariant emerges.
New geometric methods solve a conjecture for hyperbolic 3-manifolds.
problem Verifying the 1-loop conjecture for hyperbolic 3-manifolds.
method Constructing geometric ideal triangulations and solving gluing equations.
result Proves the 1-loop conjecture for a large class of hyperbolic 3-manifolds.
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
The article confirms Thurston's conjecture for a specific class of 3-manifolds using combinatorial Ricci flow.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow (CRF) with specific conditions and techniques to handle intrinsic difficulties.
result A class of 3-manifolds admits a unique complete hyperbolic metric with totally geodesic boundary.
Geometric methods prove exponential growth in continued fractions.
problem Exponential growth in partial quotients of continued fractions.
method Geometric representation of continued fractions and orbifold triangulations.
result Eventually periodic continued fractions have exponentially growing partial quotients.
The paper proves a combinatorial Ricci flow converges to hyperbolic structures on certain 3-manifolds.
problem Proving convergence of combinatorial Ricci flow to hyperbolic structures.
method Combinatorial Ricci flow on closed pseudo 3-manifolds with specific edge valences.
result Existence and uniqueness of a complete hyperbolic metric with totally geodesic boundary.
Paper presents a method to create tight triangulations of manifolds.
problem Finding tight triangulations of manifolds in higher dimensions.
method Combinatorial scheme to generate tight triangulations.
result New examples of tight triangulations in dimensions 3, 4, and 5.
Study on non-peripheral ideal decompositions of alternating knots.
problem Understanding ideal triangulations of alternating knots.
method Small cancellation properties and Dehn presentations of alternating knot groups.
result Non-peripheral ideal triangulations for all planar, reduced, alternating projections of hyperbolic knots.
Alexander's conjecture extended to infinite simplicial complexes.
problem Alexander's conjecture for infinite simplicial complexes.
method Generalization of recent result for finite simplicial complexes.
result Alexander's conjecture holds for infinite simplicial complexes.