arXiv research

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

Algorithm detects free products in disk mapping class groups.

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.

Random simple closed curves map Teichmüller space to 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.

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…

Study on homological Dehn functions of groups of type $FP_2$.

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 …

Precise computations of Dehn functions for subgroups of free group products.

Dehn quandles of groups and surfaces unify various quandle constructions.

The paper presents methods to write presentations for Dehn quandles.

Study finds the order of Dehn twists in various groups.

Study shows connectedness of Bowditch boundary persists in long Dehn fillings.

Four-dimensional Einstein Dehn filling is impossible.

Paper examines Dehn twists on non-orientable surfaces and their limitations.

Study of Dehn twists in free groups generates right-angled Artin groups.

The paper constructs Goeritz matrices from Dehn colorings.

Random hyperbolic 3-manifolds can be obtained via Dehn surgery.

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.

The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.

This paper classifies Dehn fillings of a specific 3-manifold using invariant properties.

The paper explores methods to decompose periodic maps into Dehn twists.

The paper finds minimum Dehn colors for knots and defines useful graphs for coloring.

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.

In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some …

The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.

New CAT(0) groups show superexponential subgroup Dehn functions.

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.

The paper classifies palettes of Dehn colorings for spatial graphs.

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 $L^2$-Betti numbers of Dehn fillings for special groups.

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 $\partial$-reducible. A manifold in the second family has boundary consi…

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.

New invariants distinguish spatial graphs not previously possible.

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 $n^2\log n$ 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.

Study shows Dehn twist coefficients are consistent across different actions on surfaces.