An ideal triangulation of a hyperbolic 3-manifold with one cusp is non-peripheral if no edge of is homotopic to a curve in the boundary torus of . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of . A planar project…
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48 results for “peripheral ideal”
Proves finiteness of Kauffman bracket skein modules for 3-manifolds.
problem Finiteness of Kauffman bracket skein modules for 3-manifolds.
method Constructive proof methods, alternative to Witten's original proof.
result Proves finiteness conjecture for 3-manifolds with boundary and closed 3-manifolds.
We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…
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 regular cover which has a geodesic partially truncated triangulation. The proofs us…
3-manifolds have covers with infinitely many ideal triangulations.
problem Proving the existence of infinitely many geometric ideal triangulations in certain 3-manifolds.
method Using separability of peripheral subgroups and conjugacy separability theorems.
result Every cusped hyperbolic 3-manifold has a cover with infinitely many geometric ideal triangulations.
In this paper we provide a new obstruction to 0-concordance of knotted surfaces in in terms of Alexander ideals. We use this to prove the existence of infinitely many linearly independent 0-concordance classes and to provide the first proof that the submonoid of 2-knots is not a group. The main result is that the…
By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that th…
New peripheral structure for core groups detects noninvertible knots.
problem Detecting noninvertible knots and links.
method Introduced a new peripheral structure for core groups.
result The new structure detects noninvertibility of some knots and links.
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.
Constructs the medial quandle from link's peripheral structure.
problem Building the medial quandle of links.
method From reduced Alexander module's peripheral structure.
result Medial quandle constructed successfully.
Corrects a paper on Alexander modules and answers a related question.
problem Peripheral elements in reduced Alexander modules
method Analyzes and corrects a paper on Alexander modules
result Answers a question and corrects a minor error in the original paper
The paper characterizes groups acting on real projective spaces.
problem Understanding groups acting on convex domains in real projective geometry.
method Proves structure theorem for relatively hyperbolic groups in real projective spaces.
result Characterizes groups in terms of invariant convex subsets.
A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group we define a peripheral filling procedure, which produces quotients of by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold on the fundamental group .…
Let be a cusped hyperbolic -manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in (up to conjugation) is of complex dimension the number of cusps near the hyperbolic representation. It seems natural to ask whether some …
Locally connected boundaries proven for relatively hyperbolic groups.
problem Proving local connectedness of boundaries for relatively hyperbolic groups.
method Using a group pair that is relatively one ended, and removing restrictions on cardinality and peripheral subgroups.
result The Bowditch boundary of is locally connected.
The reduced peripheral system was introduced by Milnor in the fifties for the study of links up to link-homotopy, i.e. up to isotopies and crossing changes within each link component. However, for four or more components, this invariant does not yield a complete link-homotopy invariant. This paper provides two characte…
Study shows connectedness of Bowditch boundary persists in long Dehn fillings.
problem Persistence of Bowditch boundary connectedness in Dehn fillings.
method Analysis of relatively hyperbolic group pairs and peripheral subgroups.
result Connectedness of Bowditch boundary persists in sufficiently long Dehn fillings without needing restrictions.
We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible -manifold . We shall focus on the relation between this problem and the existence of acylindrical splittings of as an amalgamated product or HNN-extension along the abeli…
Groups with semistable peripheral subgroups are semistable.
problem Semistability of fundamental groups in relatively hyperbolic groups.
method Generalization of semistability from 1-ended subgroups to finitely generated subgroups with semistable fundamental groups.
result Semistability of fundamental groups in more general relatively hyperbolic groups.
Joyce has shown that the fundamental quandle of a classical knot can be derived from consideration of the fundamental group and the peripheral structure of the knot, and also that the group and much of the peripheral structure can be recovered from the quandle. We generalize these results to arbitrary dimensions, and a…
By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…
We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The volume is still an important ingredient in this paper. Unfortunately at this poi…
We give examples of closed, oriented 3-manifolds whose fundamental groups are not isomorphic, but yet have the same sets of finite quotient groups; hence the same profinite completions. We also give examples of compact, oriented 3-manifolds with non-empty boundaries whose fundamental groups though isomorphic have disti…
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
The paper studies how Kleinian groups can be deformed while preserving their peripheral structures.
problem Determining the extent of the island of discrete representations around the identity map in the space of representations.
method By cutting up the conformal boundary of a hyperbolic 3-manifold into a fundamental domain, the paper provides a computable region within which the fundamental domain is valid, ensuring peripheral structures remain similar under small deformations.
result The paper identifies a region in the space of representations of the fundamental group of a geometrically finite manifold, showing that groups in this region have peripheral structures that look coarsely similar.
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are -holonomic, that is, they satisfy linear -difference equations with coefficients Laurent polynomials in and . We show from first principles that -holonomic sequence…
Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.
problem Boundary criterion for relative cubulation in non-one-ended subgroups.
method Showed that if boundary criterion is satisfied for a relatively hyperbolic group, the group admits a relatively geometric action on a CAT(0) cube complex.
result The refinement of the boundary criterion is useful for constructing new relative cubulations.
The problem of portfolio optimization is one of the most important issues in asset management. This paper proposes a new dynamic portfolio strategy based on the time-varying structures of MST networks in Chinese stock markets, where the market condition is further considered when using the optimal portfolios for invest…
Characterizes components of representations space for punctured surfaces.
problem Characterizing connected components of representations space.
method Using relative Euler classes, signs of peripheral elements, and generalized Milnor-Wood inequality.
result Counted total number of connected components of type-preserving representations.
Defines new representations for hyperbolic groups, unifying existing definitions.
problem Geometrically finite behavior in higher rank groups.
method Introduces a new family of discrete representations for relatively hyperbolic groups.
result Stability of these representations under certain deformations.
We prove that every non-positively curved locally symmetric manifold M of finite volume contains a compact set K such that no periodic maximal flat can be homotoped out of K.
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion m…
New equivalence relation for links using cut-diagrams.
problem Classical link concordance.
method Cut-diagrams and cut-concordance.
result Nilpotent peripheral system invariant of cut-concordance.
We give an alternative definition of relative hyperbolicity based on properties of closest-point projections on peripheral subgroups. We also derive a distance formula for relatively hyperbolic groups, similar to the one for mapping class groups.
We found a unique 4D plane that can't be simplified.
problem Finding an irreducible embedded projective plane in .
method Constructing a specific geometric shape in 4D space.
result The constructed projective plane is irreducible.
3-manifolds with toral boundary are uniquely determined by their profinite completions.
problem Determining the homeomorphism type of 3-manifolds based on their profinite completions.
method JSJ-decomposition and profinite completions of fundamental groups.
result Profinitely almost rigid 3-manifolds have finitely many homeomorphism types.
3D hyperbolic manifolds map one-to-one to their boundary character varieties.
problem Mapping 3D hyperbolic manifolds to their boundary character varieties.
method Using Bonahon-Schläfli formula and volume rigidity of discrete co-compact representations.
result Peripheral map is a birational isomorphism with its image.
Knot Group Epimorphismsmath.GT
Any knot group is the image of the group of a prime knot by a homomorphism that preserves peripheral structure. In fact, there are infinitely many such prime knots. A related partial order on knots is defined, and its properties are discussed.
Circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfacesmath.GT
Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.
problem Characterize Teichmüller spaces of hyperbolic cone surfaces.
method Construct circular foliations and shear-radius coordinates on Teichmüller spaces of hyperbolic cone surfaces.
result Shear-radius coordinates provide global coordinates on Teichmüller spaces and converge to specific metrics.
We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.
Finite index subgroups of relatively hyperbolic groups have equal index.
problem Finite index subgroups of relatively hyperbolic groups have equal index.
method Demonstrating that the number of simplices in a simplicial classifying space grows linearly with index.
result Finite index subgroups of relatively hyperbolic groups have equal index.
The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
Proves conditions for minimal surfaces in complex hyperbolic space.
problem Conditions for finite energy equivariant minimal surfaces in complex hyperbolic space.
method Analyzes peripheral holonomy and uses Higgs bundles.
result Explicit parametrization and construction of minimal surfaces.
Study measures complexity of surfaces using a new graph to prove group properties.
problem Understanding the complexity and structure of mapping class groups.
method Introduces a non-peripheral curve graph and uses it to analyze the structure of mapping class groups.
result Proves properties of the mapping class group based on the complexity measure.
Holomorphic curves in moduli spaces are quasi-isometrically immersed.
problem Understanding the geometric properties of holomorphic curves in moduli spaces.
method Analyzing the quasi-isometric immersion of holomorphic maps from hyperbolic surfaces to moduli spaces.
result Holomorphic curves are quasi-isometrically immersed with parameters depending on surface and moduli space properties.
If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .