Characterizes groups for specific types of representations into SL(d,R).
problem Understanding groups admitting certain types of representations into SL(d,R).
method Characterization and bounds on cohomological dimension.
result Bounds on cohomological dimension and characterizations of representations.
We classify SL(2;C)-representations of a Brieskorn homology 3-sphere. We show any irreducible representation into SL(2;C) is conjugate to that into either SU(2) or SL(2;R). We also give a construction of SL(2;R)-representations for a Brieskorn homology 3-sphere from PSL(2;R)-representations of the base orbifold fundame…
Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among al…
Method determines trace-free SL(2,C) reps for arborescent links.
problem Classifying trace-free representations of arborescent links.
method Completely determines trace-free mSL(2,C)-representations for arborescent links. result Concrete computations for a class of 3-bridge arborescent links.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
Study local structure of knot group representations into SL(n,C).
problem Understanding the local structure of knot group representations.
method Analysis of tangent cone and use of Luna's slice theorem.
result Local structure of representation variety at diagonal representations.
Anosov representations uniquely determined by boundary maps for surface groups to SL(3,R).
problem Uniqueness of Anosov representations determined by boundary maps.
method Analysis of boundary maps and representation properties.
result Anosov representations uniquely determined by boundary maps if not reducible.
We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy class…
Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given…
Researchers create a new invariant for knot theory.
problem Constructing invariants for knot theory.
method Extending a result to all irreducible representations of sl3. result Existence of a new invariant FKsl3 for any positive braid knot K. Paper studies complex Lagrangian surfaces and their relation to SL(3,C)-representations.
problem Minimal Lagrangian surfaces in bi-complex hyperbolic space and their representations.
method Introduces bi-complex Higgs bundles and parameterizes SL(3,C)-quasi-Fuchsian representations. result Parameterization of SL(3,C)-quasi-Fuchsian representations by an open set in Teichmüller space. Let Mφ be a surface bundle over a circle with monodromy φ:S→S. We study deformations of certain reducible representations of π1(Mφ) into SL(n,C), obtained by composing a reducible representation into SL(2,C) with the irreducible representation $\text{SL}(2,\mathb…
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Uti…
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exception…
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
Study of ω-Borel invariant for representations into SL(n,Cω).
problem Characterizing representations into SL(n,Cω) using ω-Borel invariant. method Defined ω-Borel invariant βnω(ρω) for representations ρω:ΓightarrowSL(n,Cω), studied sequences of ω-bounded representations and their limits. result If a sequence of representations ρl into SL(2,C) determines a reducible action on the asymptotic cone Cω(H3,d/λl,O), then β2ω(ρω)=0. The aim of this article is to study the existence of certain reducible, metabelian representations of knot groups into SL(n,C) which generalise the representations studied previously by G.~Burde and G.~de Rham. Under specific hypotheses we prove the existence of irreducible deformations of such repr…
We determine the characters of SL(2) representations of groups and surface groups.
New examples show embeddings not approximated by Anosov representations.
problem Understanding quasi-isometric embeddings of word hyperbolic groups into SL(d,R). method Constructing specific examples of embeddings that are not limits of Anosov representations.
result Analogous density theorem does not hold for SL(d,R) when d⩾5. Generic Hitchin representations generate dense subgroups.
problem Understanding dense subgroups in SL_n(R) representations.
method Using a theorem by Rapinchuk, Benyash-Krivetz, and Chernousov.
result Generic Hitchin representations are strongly dense.
Unified framework counts knot representations into SU(2) and SL(2,R).
problem Counting knot representations into SL(2,R) with fixed holonomy.
method Unified framework using geometric transition and character varieties.
result SL(2,R) count determined by SU(2) count and integer h(K).
Extends Lawrence's representations to integral Uqsl(2) Verma-modules and braid groups.
problem Integrating Lawrence's representations into Uqsl(2) Verma-modules and braid groups. method Defining homological operators and showing they provide a representation for Uqsl(2), establishing isomorphisms and preserving key properties. result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
Study topological components of surface group representations into SL(2,R) and PSL(2,R).
problem Understanding the structure of representation spaces of surface groups.
method Utilized the signature formula to determine connected components of representation spaces.
result Determined the number of connected components for different types of holonomies.
Given a knot K in an integral homology sphere with exterior N_K, there is a natural action of the cyclic group Z/n on the space of SL(n,C) representations of the knot group π_1(N_K), and this induces an action on the SL(n,C) character variety. We identify the fixed points of this action in terms of characters of metabe…
New computations show sl(N) homology is related to SU(N) representations of knots.
problem Computing colored sl(N) homology for nontrivial knots and links.
method Using SU(N) representations of knot complements, we compute homology and show isomorphisms.
result Colored sl(N) homology is isomorphic to the cohomology of SU(N) representations of knot complements.
High-dimensional components found in knot group representations.
problem Characterizing high-dimensional components in knot group representations.
method Analyzing (n−1)-dimensional components and proving existence for sufficiently large n. result Existence of high-dimensional components in SL(n,C)-character varieties for hyperbolic knots. Deforms surface groups to be Zariski dense in SL(n,R)
problem Finding Zariski dense surface groups in SL(n,R)
method Deforming K-integral representations of surface groups result Generalizes Long and Thistlethwaite's method to SL(n,R)
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.
We observe the twisted Alexander polynomial for metabelian representations of knot groups into SL(2,C) and study relations to the characterizations of metabelian representations in the character varieties. We give a factorization of the twisted Alexander polynomial for irreducible metabelian representations with the ad…
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.
Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is conjugate to an SU(n) representation. In this case we show it determines a smoot…
We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the charac…
The paper proves a criterion for L-space knots and their representations.
problem Conditions for abelian SL(2,R)-representations of knot groups. method Continuous family of irreducible representations converging to abelian representations.
result Alexander polynomial of nontrivial L-space knots has odd order on the unit circle.
Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.
problem Characterizing SL(2,R) representations on a once-punctured torus.
method Introduction of spectrum as a subset of projective measured laminations, analysis of dynamics of cocycles.
result Spectrum of a generic representation on a once-punctured torus is a Cantor set.
We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of 3-manifolds to SL(4,R) that factor over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) = S0(3,1). We also discuss consequences for the number of connected components of SL(4,R)-character…
Let K be a knot in S3 and X its complement. We study deformations of non-abelian, metabelian, reducible representations of the knot group π_1(X) into SL(n,C) which are associated to a simple root of the Alexander polynomial. We prove that certain of these metabelian reducible representatio…
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
We prove Riley's conjecture on the number of parabolic SL(2,R) representations of 2-bridge knot groups.
New braid group action defined on projective quantum sl(2) modules.
problem Defining a new braid group action on quantum sl(2) modules.
method Action via R-matrix on tensor powers of simple projective modules.
result The action is faithful for the extended representation.
Surface groups have solvable representations in SL(2,R).
problem Representing surface groups without simple loops.
method Torsion-free group of upper-triangular matrices in SL(2,R).
result No simple loop in the kernel of representations.
We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irred…
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.
The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.
problem Asymptotic behavior of colored Jones polynomial for the figure-eight knot.
method Analyzing the polynomial's behavior as N approaches infinity and evaluating it at specific points.
result The polynomial corresponds to an SL(2;R) representation of the knot complement.
Let q be a 2Nth root of unity where N is odd. Let Uq(sl2) denote the quantum group with large center corresponding to the lie algebra sl2 with generators E,F,K, and K−1. A semicyclic representation of Uq(sl2) is an N-dimensional irreducible representation $ρ:U_q(sl_2)\rightarrow M_N(\mathbb{C}…
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
Quantum invariants derived from Uq(sl2) link holonomy.
problem Quantum invariants of links and their relations.
method Using quantum groups and Schur-Weyl duality, constructing quantum holonomy invariants.
result Quantum invariants of links can be derived from Uq(sl2) representations.