Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.
problem Understanding the structure of quantized SL2-character variety of a once-punctured torus.
method Analyzing the quantized algebra and its subalgebras isomorphic to Coulomb branches.
result Three Z2-invariant subalgebras of the quantized algebra are isomorphic to Coulomb branches. We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
Skein modules over 3-manifolds are shown to form line bundles.
problem Understanding the structure of skein modules over 3-manifolds.
method Using the Frobenius morphism, the skein module is mapped to a coherent sheaf over the SL2 character scheme.
result When the character scheme is reduced, the sheaf is a line bundle.
Study of non-Archimedean Hitchin map for SL2(F) characters.
problem Characterizing representations of SL2(F) characters.
method Equivariant harmonic maps into R-trees, Jenkins-Strebel differentials.
result The non-Archimedean Hitchin map is continuous and its image is contained in Jenkins-Strebel differentials.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
The paper proves a vanishing identity for twist knots using character varieties.
problem Proving a vanishing identity for adjoint Reidemeister torsions of twist knots.
method Using Jacobi's residue theorem on the SL2(C)-character variety.
result The conjecture about vanishing identities for twist knots is proven.
The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.
problem Defining and analyzing the adjoint Reidemeister torsion for connected sums of knots.
method Defined a natural way to compute the adjoint Reidemeister torsion for high-dimensional components of the character variety.
result The adjoint Reidemeister torsion is locally constant and satisfies the vanishing identity.
Study multiplicity of non-acyclic SL2-representations and L-functions of Whitehead links.
problem Understanding the multiplicity of non-acyclic SL2-representations and their L-functions.
method Geometric interpretation of Reidemeister torsion divisors and application to L-functions.
result Prove multiplicity two for odd-twisted Whitehead links.
We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming para…
Study non-acyclic SL2-representations of twist knots and their L-functions.
problem Characterize SL2-representations of twist knots and their properties.
method Character variety, Reidemeister torsion, Chebyshev polynomials, and L-functions.
result Non-acyclic SL2-representations lie on the line x=y in character variety, and their orders are related to (-3)-Dehn surgery.
We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it admits no non-trivial cyclic or finite surgeries. We also summarize similar results f…
Character scheme of small Seifert 3-manifolds is reduced if and only if no exceptional abelian character exists.
problem Character scheme of small Seifert 3-manifolds and its reduction condition.
method Complete description of the SL2(C)-character scheme of small Seifert 3-manifolds.
result Character scheme is reduced if and only if no exceptional abelian character exists, and exceptional abelian character has multiplicity 2.
The paper defines a new algebra structure on skein modules of 3-manifolds and shows it's isomorphic to a known algebra.
problem Defining and studying algebraic structures on skein modules of 3-manifolds.
method Introducing and analyzing a new algebra structure K±i0(M) based on links in 3-manifolds. result The algebra K±i0(M) is naturally isomorphic to a known algebra when the parameter is ±1. Paper extends algebraic geometry results to hyperbolic link complements.
problem Understanding algebraic and number-theoretic properties of canonical curves.
method Generalizes Chinburg-Reid-Stover's results to hyperbolic link complements.
result Azumaya algebra does not extend to canonical surfaces.
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.
Arithmetic study of knots connects homology and SL2 representations.
problem Understanding cyclic covers of knots and their SL2 representations.
method Analyzing cyclic branched covers and using SL2 representations.
result Liminal SL2 representations exist for certain knots.
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
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. 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.
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.
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…
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.
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.
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.
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…
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.
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.
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.
We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.
Let Γ be a finitely generated group and G a real form of SLn(C). We propose a definition for the G-character variety of Γ as a subset of the SLn(C)-character variety of Γ. We consider two anti-holomorphic involutions of the SLn(C) character variet…
Minimal action of mapping class group on character variety.
problem Character variety of Deroin-Tholozan representations.
method Geometric perspective using symplectic structure.
result Infinite mapping class group orbits are dense.
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.
In this paper we consider some families of links, including (-2,2m+1,2n)-pretzel links and twisted Whitehead links. We calculate the character varieties of these families, and determine the number of irreducible components of these character varieties.
We describe a birational map between subvarieties in the character varieties of mutative 3-manifolds. By studying the birational map, one can decide in certain circumstances whether a mutation surface is detected by an ideal point of the character variety.
We will construct twisted versions of the wild character varieties.
The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.
problem Compactifying character varieties of punctured surfaces.
method Projective compactifications using ideal triangulations and Komyo's method.
result Boundary divisors are toric varieties and the boundary complex is a sphere.
Study shows how certain knots and tori are detected by ideal points in character varieties.
problem Detecting essential twice-punctured tori in character varieties.
method Extending Tillmann's limiting character method to new families of knots and tori.
result Explicit determination of limiting characters at ideal points for specific knots and tori.
Study of twisted L2-torsion on 3-manifold character varieties.
problem Analyzing L2-torsion on character varieties of 3-manifolds. method Definition and study of twisted L2-torsion on character varieties, proving analyticity in hyperbolic cases. result Twisted L2-torsion is a real analytic function in a neighborhood of holonomy representations of hyperbolic 3-manifolds. Authors determine the SL(2,C) character variety of a specific knot without computer aid.
problem Determine the SL(2,C) character variety of a specific knot without computational assistance.
method Develop an efficient method for working with conjugacy classes of four elements of SL(2,C).
result Determine the character variety of the knot 8_18 efficiently and software-free.
Compactifies character varieties for group actions.
problem Compactify character varieties for better topological analysis.
method Real spectrum compactification of character varieties in semisimple Lie groups.
result Provides a compactification with good topological properties.
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. We show that for any knot there exist only finitely many irreducible metabelian characters in the SL(2,C)-character variety of the knot group, and the number is given explicitly by using the determinant of the knot. Then it turns out that for any 2-bridge knot a section of the SL(2,C)-character va…
We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the SL2(C) character variety, and following Boyer and Zhang in the PSL2(C) character variety as well. We show that any slope detected on a canoni…
Researchers describe the dual of cohomology generators for SU(2) character varieties of surfaces.
problem Understanding the cohomology structure of SU(2) character varieties of surfaces.
method Explicit description of Poincaré duals of cohomology generators.
result An explicit description of the Poincaré dual of each generator of the rational cohomology ring.