New symplectic instanton homology constructed via traceless character varieties.
problem Constructing symplectic instanton homology for 3-manifolds.
method Using traceless character varieties to define a Lagrangian Floer invariant.
result Symplectic instanton homology conjecturally equivalent to a gauge theoretic variant.
Study character varieties of tangles to map immersed curves in the pillowcase.
problem Characterizing holonomy-perturbed traceless SU(2) character varieties.
method Examining marked tangles as endomorphisms in the cobordism category and using holonomy-perturbed traceless character variety functor.
result Endomorphisms of immersed curves in the pillowcase have the same image.
Character variety of a 6-punctured sphere mapped to CP^3 with specific branch points.
problem Characterizing the traceless SU(2) character variety of a 6-punctured 2-sphere.
method Analysis of 2-fold branched cover and application of Narasimhan-Ramanan theorem.
result Character variety corresponds to a 2-fold branched cover of CP^3 over a singular Kummer surface.
New method to compute tangle sums in character varieties.
problem Computing character varieties for tangle sums.
method Cut-and-paste methods to construct perturbed character varieties.
result Bounding cochains can be used to recover differential of Iatural. Given a 2-stranded tangle in a $\ZZ/2$ homology ball, T⊂Y, we investigate the character variety R(Y,T) of conjugacy classes of traceless SU(2) representations of π1(Y∖T). In particular we completely determine the subspace of binary dihedral representations, and identify all of R(Y,T) for many t…
Holonomy perturbations affect character varieties of tangles, leading to smooth manifolds.
problem Character varieties of tangles under holonomy perturbations.
method Stratified transversality arguments and holonomy perturbations along curves in cylinders.
result Holonomy perturbations yield smooth manifolds without boundary.
Study tangle invariants using pillowcase Fukaya category and character varieties.
problem Tangle invariants and their relationships with Khovanov cohomology.
method Define a twisted complex of compact Lagrangians in the pillowcase's Fukaya category, show functoriality, and relate to Khovanov cohomology.
result Hom set with a specific Lagrangian is naturally isomorphic to reduced Khovanov chain complex.
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. Study shows how tangle moduli spaces relate to boundary surfaces.
problem Relating tangle moduli spaces to boundary surfaces.
method Use of composition in the Weinstein category and SU(2) holonomy perturbations.
result Holonomy perturbed SU(2) traceless flat moduli space is a Lagrangian immersion.
The paper studies traceless SU(2) representations and instanton gradings for specific knot types.
problem Analyzing traceless SU(2) representations and instanton gradings for two-bridge and (3,n)-torus knots. method Representation-theoretic data assembly and verification; explicit computation of character varieties and spectral-flow gradings.
result Explicit characterization of traceless representations and instanton gradings for specific knot types.
We define an elementary relatively Z/4 graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless SU(2) character varieties of 2-tangle decompositions of knots. Calculations for torus knots…
Scheme for generating sets in knot homology for lens spaces.
problem Constructing generating sets for knot homology in lens spaces.
method Heegaard splitting and Lagrangian intersections in character varieties.
result Illustrated construction of generating sets for several example knots.
Study shows how tangle geometry maps onto pillowcase surfaces.
problem Understanding geometric structures on tangles and their boundaries.
method Analyzes the traceless SU(2) flat moduli space of the earring tangle and its boundary.
result The restriction map from the surface to the pillowcases is a Lagrangian immersion.
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
New knot homology theory from symplectic geometry.
problem Developing a symplectic counterpart to instanton knot homology.
method Using symplectic character varieties and Floer homology, a new invariant for knots in 3-manifolds is constructed.
result Symplectic instanton knot homology (SIK) is a new invariant of knots and links in 3-manifolds.
The paper develops a new homology theory for 3-manifolds and proves its functoriality.
problem Developing a new homology theory for 3-manifolds.
method Using symplectic geometry and Floer theory, constructing a new homology theory and proving its functoriality.
result The symplectic instanton homology is functorial with respect to cobordisms.
New twisted versions of wild character varieties created.
problem Constructing twisted wild character varieties.
method Creating new versions of wild character varieties.
result Twisted versions of wild character varieties constructed.
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
Quantum character varieties unify four construction methods.
problem No specific problem stated; unification of approaches.
method Four different approaches to construction.
result Unified understanding of quantum character varieties.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
Character varieties of 2-bridge knots and links explained using Farey recursion.
problem Understanding character varieties of 2-bridge knots and links.
method Using Farey recursion to define polynomials for character varieties.
result Character varieties described in terms of polynomials defined by Farey recursion.
Computes dimensions of representation and character varieties for 2 and 3-dimensional orbifolds.
problem Computing dimensions of representation and character varieties for orbifolds.
method Analyzes varieties of representations and characters of hyperbolic and non-hyperbolic orbifolds, provides tools for computation.
result Shows that the dimension of a specific component of characters equals half the dimension of the variety of characters of the boundary.
Researchers describe character varieties for Hopf links, proving geometric properties.
problem Character variety geometry of Hopf links with n twists. method Geometric descriptions of irreducible and totally reducible representations.
result Complete geometric description of SU(2)-character variety for r=2. Study character varieties for real forms of SL_n(C).
problem Characterize representations of groups in real forms of SL_n(C).
method Define and analyze G-character varieties, study involutions, and examples.
result Irreducible representations fixed by involutions are conjugate to real forms.
Detects essential surfaces in knots using character variety intersections.
problem Identifying essential surfaces in knot theory.
method Analyzing intersections in the character variety of hyperbolic knots.
result Intersection points in the character variety detect Seifert surfaces.
Character variety of Whitehead link described in detail.
problem Character variety of Whitehead link.
method Nice description of a Zariski open subset.
result Zariski open subset of the character variety described.
Efficient method computes mSL(3,C)-character varieties for two-generator groups.
problem Computing mSL(3,C)-character varieties for two-generator groups. method Efficient method for two-generator groups.
result Efficient computation of mSL(3,C)-character varieties. Proves Torelli group action is ergodic on Lie group character varieties.
problem Ergodicity of Torelli group action on compact character varieties.
method Proof based on properties of connected, semi-simple Lie groups.
result Torelli group action on compact character varieties is ergodic.
Study character varieties of odd classical pretzel knots.
problem Determine mSL(2,C)-character varieties for odd classical pretzel knots. method Compute A-polynomial for each knot.
result Character varieties of odd classical pretzel knots computed.
Study SL2(F)-character varieties of knots over fields of odd characteristic.
problem Character varieties of knots over fields of odd characteristic exhibit ramification phenomena.
method Investigate sufficient conditions for ramification using double branched covers and Alexander polynomials.
result Character varieties of knots over fields of odd characteristic can present ramification phenomena.
The paper extends character varieties to marked surfaces and discovers their triangular decompositions.
problem Character varieties on marked surfaces and their properties.
method Triangulations of marked surfaces and cohomological groups.
result Stated character varieties admit triangular decompositions.
We study some basic properties of the variety of characters in PSL(2,C) of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to SL(2,C). We also study t…
Detecting essential surfaces in 3-manifolds and orbifolds using character varieties.
problem Detecting essential surfaces in 3-manifolds and orbifolds.
method Extending Culler and Shalen's construction to 3-orbifolds using SL2(C) character variety. result Any slope detected on a canonical component of the (P)SL2(C) character variety of a one-cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Study character varieties in SL_2 and Kauffman skein algebras for surfaces and 3-manifolds.
problem Understanding the algebraic geometry of character varieties.
method Focus on surface and 3-manifolds groups, construct Reidemeister torsion as a rational volume form.
result Construct the Reidemeister torsion as a rational volume form on the character variety.
Finite automorphisms of SL(2,C) character varieties identified.
problem Character varieties of surfaces with negative Euler characteristic.
method Algebraic automorphisms and mapping class groups studied.
result Finite extension of mapping class group identified.
Study character varieties of hyperbolic 3-manifolds using bundle methods.
problem Character varieties of hyperbolic 3-manifolds in once-punctured torus bundles.
method Restrict characters to the fibre and analyze branched covering maps.
result Infinite family of hyperbolic once-punctured bundles with unbounded genus.
A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.
Defines Reidemeister torsion form on 3-manifold character varieties.
problem Character variety singularities and vanishing torsion.
method Differential form on character variety, Alexander polynomial, Culler-Shalen theory.
result Torsion vanishes only at singular points, related to reducibility and Euler characteristic.
Study of pseudo-Anosov actions on SU(2)-character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on SU(2)-character variety. method Analysis of mapping class group subgroups and their pseudo-Anosov elements.
result Existence of a subgroup containing infinitely many pseudo-Anosov elements with invariant rational functions.
Study of SU(2,1) character varieties on one-holed torus.
problem Characterize representations of mapping class group on SU(2,1) character variety.
method Explicit description of SU(2,1) character variety, use of Farey graph adaptation, and mapping class group action analysis.
result Description of an open domain of discontinuity for mapping class group action.
The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields…
Character varieties of knot groups into SU(3) are stratified and analyzed.
problem Characterizing representations of knot groups into SU(3).
method Stratification of character variety into reducible, 2D+1D, and irreducible representations.
result Homotopy equivalence between SU(3) and SL(3,C) character varieties.
Study character varieties of even pretzel knots, determining their mSL(2,C)-representations.
problem Determine character varieties of even classical pretzel knots.
method Compute irreducible mSL(2,C)-representations of even classical pretzel knots. result Clarify the steps to compute the A-polynomial of even classical pretzel knots.
New compactification for character varieties with good topological properties.
problem Compactification of character varieties with good topological properties.
method Announced a new compactification with interpretations of ideal points.
result Relates to Weyl chamber length compactification and applies to maximal and Hitchin representations.
Study character varieties of arborescent knots and hyperbolic knots.
problem Computing and understanding mSL(2,C)-character varieties of knots. method Described a procedure for computing mSL(2,C)-character varieties of arborescent knots, clarified facts about arborescent tangles, and studied character varieties of hyperbolic knots. result Found that each hyperbolic knot with essential surfaces has a 1-dimensional character variety.
Study character variety of a special 3-manifold, finding a polynomial formula.
problem Character variety of a specific 3-manifold.
method Determined irreducible SL(2,C) character variety, derived twisted Alexander polynomial.
result Formula for twisted Alexander polynomial of SL(2,C) representations.
Anosov representations of surface groups are complex manifolds.
problem Characterizing the geometry of Anosov representations of surface groups.
method Analyzing the character varieties of Anosov representations into $\SL(n , \C)$.
result Character varieties of Anosov representations are complex manifolds of specific dimension.