Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.
problem Analyzing actions of Dehn twists in geometric group theory.
method Application of Dynnikov coordinates to describe orbits and dynamics of Dehn twists in a thrice-punctured disc.
result The action of Dehn twists has a geometric meaning as a piecewise linear Z2-automorphism. Algorithm detects free products in disk mapping class groups.
problem Detecting free products in mapping class groups of punctured disks.
method Algorithm based on Dynnikov coordinates to verify completeness and reveal free product structure.
result Algorithm determines exact structure of free products generated by Dehn twists.
We define Ptolemy coordinates for representations that are not necessarily boundary-unipotent. This gives rise to a new algorithm for computing the SL(2,C) A-polynomial, and more generally the SL(n,C) A-varieties. We also give a formula for the Dehn invariant of an SL(n,C)-representation.
Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.
problem Studying representations of Kauffman bracket skein algebras at roots of unity.
method Using the action of the skein algebra on the skein module of the handlebody.
result Explicit reconstruction of unique representation with fixed classical shadow.
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially use the concept of a simple earthquake which is a particular case of a Fenchel-Nie…
Let Σ be a compact, orientable surface of negative Euler characteristic, and let h be a complete hyperbolic metric on Σ. A geodesic curve γ in Σ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word …
Study on homological Dehn functions of groups of type FP2.
problem Understanding the homological Dehn functions of groups of type FP2. method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type FP2 with quartic homological Dehn function and unsolvable word problem. Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
Dehn quandles of groups and surfaces unify various quandle constructions.
problem Understanding and unifying various quandle constructions.
method Introducing Dehn quandles of groups and subsets, proving properties and embeddings.
result Dehn quandles embed naturally into their enveloping groups, and enveloping groups of certain quandles are the quandles themselves.
The paper presents methods to write presentations for Dehn quandles.
problem Writing explicit presentations for Dehn quandles.
method Two approaches: for Garside and Gaussian groups, and for general Dehn quandles with known centralisers.
result Examples of Dehn quandles including spherical Artin groups, surface groups, and mapping class groups.
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
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.
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. Study of Dehn twists in free groups generates right-angled Artin groups.
problem Understanding dynamics of Dehn twists in free groups.
method Geometry of spheres, tori, and curves in a doubled handlebody; analysis of compatibility conditions.
result Sufficiently large powers of Dehn twists generate right-angled Artin groups.
Study connects surface twists to curve invariants.
problem Understanding fractional Dehn twists on Riemann surfaces.
method Characterization of pseudo-periodic maps and bounds on fractional twists.
result Established relationship between twists and curve invariants.
The paper constructs Goeritz matrices from Dehn colorings.
problem Constructing Goeritz matrices from Dehn colorings.
method Purely algebraic construction of Goeritz matrices from Dehn coloring matrices for prime knot diagrams.
result A new method to construct Goeritz matrices from Dehn colorings.
Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
problem Understanding the generic properties of hyperbolic 3-manifolds.
method Counting model on links and Dehn surgeries.
result Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n. If a simple 3-manifold M admits a reducible and a toroidal Dehn filling, the distance between the filling slopes is known to be bounded by three. In this paper, we classify all manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3.
This paper classifies Dehn fillings of a specific 3-manifold using invariant properties.
problem Classifying Dehn fillings of a 2-cusped hyperbolic 3-manifold.
method Using the product of derivatives of holonomies of core geodesics as an invariant.
result Dehn fillings with sufficiently large coefficients are classified.
The paper explores methods to decompose periodic maps into Dehn twists.
problem Factoring periodic maps into Dehn twists.
method Various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy.
result Existence of conjugates of periodic maps whose product is pseudo-Anosov.
The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.
problem Finding the minimum number of colors for Dehn colorings of knots.
method Analyzes Dehn colorings for knots and defines R-palette graphs. result For Dehn p-colorable knots, the minimum number of colors is at least ⌊log2pfloor+2. Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…
Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.
problem Distinguishing homotopy types of Gromov-Thurston manifolds
method Using virtual Dehn fillings of relatively hyperbolic groups
result Criteria to distinguish homotopy types
The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.
problem Determining the minimum number of Dehn colors for knots.
method Using symmetric local biquandle cocycle invariants to evaluate minimum Dehn colors.
result There exist knots distinguished by minimum numbers of Dehn colors.
New CAT(0) groups show superexponential subgroup Dehn functions.
problem Understanding subgroups of CAT(0) groups with superexponential Dehn functions.
method Construction of specific 4- and 6-dimensional CAT(0) groups.
result Found subgroups with Dehn functions exp(n)(xm) and exp(n)(xα). We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function a…
Study shows arithmetic properties of specific hyperbolic Dehn fillings.
problem Arithmeticity of one-cusped Dehn fillings of specific link complements.
method Investigation of cusp fields, trace fields, and invariant trace fields.
result No one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic.
The paper classifies palettes of Dehn colorings for spatial graphs.
problem Classifying spatial graph diagrams using Dehn colorings.
method Examining vertex conditions and palettes for spatial graphs.
result Spatial graphs can be distinguished by the number of Dehn colorings with specific palettes.
We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients.
It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.
Study L2-Betti numbers of Dehn fillings for special groups.
problem Investigate L2-Betti numbers of Dehn fillings. method Prove L2-Betti numbers equality for virtually special groups. result Verify Singer Conjecture for certain Einstein manifolds.
We give three infinite families of examples of nonhyperbolic Dehn fillings on hyperbolic manifolds. A manifold in the first family admits two Dehn fillings of distance two apart, one of which is toroidal and annular, and the other is reducible and ∂-reducible. A manifold in the second family has boundary consi…
Generates spin structure stabilizers using Dehn twists.
problem Determining generating sets for spin structure stabilizers.
method Using Dehn twists on specific curves.
result Explicit generating sets for stabilizers of spin structures.
Exceptional Dehn surgeries have been classified for 2-bridge knots and Montesinos knots of length at least 4. In this paper we classify all toroidal Dehn surgeries on Montesinos knots of length 3.
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proo…
Computes A-polynomials of knots from Whitehead sister link fillings.
problem Computing A-polynomials for knots from Whitehead sister link fillings.
method Using results on A-polynomials of Dehn fillings, formulas are derived for the A-polynomials of knots.
result Formulas to compute A-polynomials for knots from Whitehead sister link fillings.
New invariants distinguish spatial graphs not previously possible.
problem Distinguishing spatial graphs using Dehn colorings.
method Developed vertex-weight invariants based on Dehn colorings.
result Found spatial graphs distinguishable by vertex-weight invariants.
We construct nontrivial roots of Dehn twists about nonseparating curves.
In their article "The shape of hyperbolic Dehn surgery space," Hodgson and Kerckhoff proved a powerful theorem, half of which they used to make Thurston's Dehn surgery theorem effective. The calculations derived here use both halves of Hodgson and Kerckhoff's theorem to give bounds leading towards a practical algorithm…
The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-inte…
We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n2logn and other small Dehn functions. Some of these groups have undecidable conjugacy problem.
We use sutured manifold theory, essential laminations and essential branched surfaces to establish the upper bounds of distances between certain types of nonsimple Dehn surgery slopes. This is the revised version of an earlier preprint {\it Dehn surgery and simple manifolds.}
New group theory insights on knot surgery results.
problem Understanding non-simply connected 3-manifolds from Dehn surgery.
method Group theoretic analysis of Property P conjecture variations.
result New group theoretic perspectives on Dehn filling.