Computed Casson-Lin invariant for Hopf link.
problem Computing the Casson-Lin invariant for specific links.
method Direct computation and analysis of the Hopf link.
result Determined the sign in the linking number formula.
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48 results for “SU(2) Casson-Lin invariant”
New Casson-Lin invariants for multi-component links.
problem Counting projective SU(N) representations of link groups.
method Defined Casson-Lin invariants for links with more than one component.
result Invariants vanish for split links under mild conditions.
New invariant counts SU(2) representations for links.
problem Counting irreducible SU(2) representations for links.
method Signed count of irreducible representations with fixed traces.
result Invariant is a sum of multivariable signatures for specific links.
Note on spherical quandlesmath.GT
The paper connects knot representations and spherical quandle colorings.
problem Defining the Casson-Lin invariant for knots.
method One-to-one correspondence between -representations and spherical quandle colorings.
result Geometric interpretation of the trace-free condition for the Casson-Lin invariant.
Extends Benard-Conway invariant to all two-component links.
problem Counting irreducible SU(2) representations for two-component links.
method Counting irreducible SU(2) representations with fixed meridional traces.
result Invariant equals symmetrized multivariable link signature for (2, 2n)-torus links.
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).
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking number of the link. Our construction generalizes that of X.-S. Lin who defined a sim…
Proves generic independence and additivity of SL(2,C) Casson-Lin invariant.
problem SL(2,C) Casson-Lin invariant's parameter dependence and additivity under knot sums.
method Topology, microlocal analysis, algebraic geometry, Behrend functions.
result Generically independent and additive invariant under knot sums.
A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Torimath.GT
The paper generalizes knot signatures to tori using representations and invariants.
problem Generalizing knot signatures to tori and defining new invariants.
method Defining a signed count of irreducible representations for tori complements and relating it to known invariants.
result Defines a new invariant for tori that recovers known invariants and connects to Floer homology.
The paper connects volume conjecture with invariants and their limits.
problem Understanding the volume conjecture for invariants.
method Using symmetry properties and cyclotomic expansions, the authors propose and prove conjectural formulas for invariants.
result Proof of conjectural formulas for invariants, including volume conjecture.
The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.
problem Existence of invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.
method Decomposing Lie algebras and tangent spaces, parametrizing scalar products, and computing Ricci tensors for invariant metrics.
result Existence of invariant Einstein metrics on specific special unitary groups and complex Stiefel manifolds.
Invariant description of SU(2)-structures on 5-manifolds developed.
problem Characterizing and describing SU(2)-structures on spin 5-manifolds.
method Spinor approach to characterize subspaces inducing SU(2) isomorphism, quaternionic structure induction, and invariant derivation of covariant derivatives.
result Invariance of certain components of the covariant derivative derived.
Study coclosed G2-structures on SU(2)²-invariant manifolds.
problem Existence and classification of coclosed G2-structures on specific manifolds.
method Analysis of half-flat SU(3)-structures and boundary conditions.
result Existence of coclosed G2-structures on R⁴ × S³, no such structures on S⁴ × S³.
Study on Riemannian properties of SU_n using bi-invariant metric.
problem Properties of SU_n with a specific bi-invariant metric.
method Analyzes distance, diameter, and geodesics in SU_n.
result Parametrizes minimizing geodesic segments using a complex Grassmannian.
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
Researchers found the -invariant for is constant regardless of .
problem Exploring the -invariant for quotient groups .
method Analyzing the -invariant for and extending to .
result The -invariant for is independent of .
A perturbative SU(3) Casson invariant for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…
This paper proves a symplectic formula for SU(n) generalized Casson invariants.
problem Establishing a relationship between gauge theory and generalized Casson invariants.
method Using intersection numbers of Lagrangian submanifolds.
result Analogous identification result for SU(n) generalized Casson invariants.
Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in the context of mutant knots are also discussed.
No left-invariant hypercomplex structures found on compact Lie groups.
problem Existence of left-invariant hypercomplex structures on compact Lie groups.
method Elementary algebraic arguments to show non-existence.
result Compact Lie groups of dimension do not admit left-invariant hypercomplex structures.
Constructs instantons on a specific G_2-manifold limit.
problem Finding instantons on a particular G_2-manifold limit.
method Dynamical systems approach involving perturbations of abelian and cone solutions.
result Obtains a 1-parameter family of SU(2) instantons with bounded curvature.
New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) conne…
Study shows uniform doubling property for SU(2) geometries.
problem Estimating properties of left-invariant geometries on SU(2).
method Analyzes all left-invariant geometries on SU(2) and proves uniform doubling.
result Left-invariant geometries on SU(2) are uniformly doubling.
New invariants for 3-manifolds derived from equivariant Cerf theory.
problem Existence of perturbative Casson invariants on integer homology spheres.
method Equivariant Cerf theory for Morse functions, adapted to infinite-dimensional setting.
result Existence and explicit formula for Casson invariants.
We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…
Researchers calculate BPS invariants for Seifert manifolds using Witten-Reshetikhin-Turaev invariants.
problem Calculating BPS invariants for Seifert manifolds.
method Using exact expressions for Witten-Reshetikhin-Turaev invariants and homological blocks.
result Homological blocks for Seifert manifolds calculated from Witten-Reshetikhin-Turaev invariants.
New gradient Ricci solitons found for invariants.
problem Exploring gradient Ricci solitons with symmetry.
method Construction of new solitons using cohomogeneity one group actions.
result 3-parameter families of complete -invariant asymptotically conical expanding gradient Ricci solitons.
New steady gradient Ricci solitons found on specific four-manifolds.
problem Finding steady gradient Ricci solitons on specific four-manifolds.
method Center manifolds and topological degree theory.
result New families of complete, -invariant steady gradient Ricci solitons constructed.
Study of pseudo-Anosov actions on -character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on -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.
An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…
Study cohomology spaces of left-invariant involutive structures on SU(2).
problem Computing cohomology spaces of left-invariant involutive structures on SU(2).
method Understanding irreducible representations and left-invariant vector fields on SU(2).
result Computed smooth cohomology spaces of a corank 1 structure.
We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.
We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.
We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.
Defining an SU(3)-Casson/U(2)-Seiberg-Witten integer invariant for integral homology 3-spheresmath.DG
The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat SU(3)-connections. This obstruction is the non-trivial spectral flow of a family of…
New Einstein metrics found on SU(N) without being naturally reductive.
problem Finding non-naturally reductive Einstein metrics on SU(N).
method Using generalized flag manifolds and specific orthogonal bases of Lie subalgebras.
result Obtained new invariant Einstein metrics on SU(N).
Constructs a trace invariant for 3-manifold skein spaces.
problem Invariants for 3-manifold skein spaces.
method Local diffeomorphism invariant trace construction.
result Local diffeomorphism invariant trace on skein space.
Formula conjectured for refined SU(3) Vafa-Witten invariants of surfaces.
problem Calculating refined SU(3) Vafa-Witten invariants for smooth surfaces.
method Proved modularity transformation and used Mochizuki's formula and Maulik-Thomas's definition.
result Conjectured formula satisfies refined S-duality and verified in examples.
Researchers found the diameter of a specific type of sphere in a mathematical group.
problem Computing the diameter of a special type of sphere in a mathematical group.
method Used a left-invariant axisymmetric Riemannian metric to compute the diameter.
result Computed the diameter of Berger's sphere.
New one-parameter families of -invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with -symmetry.
method Gauge theory on asymptotically conical Calabi-Yau 3-folds with co-homogeneity one action.
result New one-parameter families of invariant instantons found.
Proved cyclotomic expansion for double twist knots' HOMFLY-PT invariants.
problem Proving cyclotomic expansion for colored HOMFLY-PT invariants of double twist knots.
method Cyclotomic expansion formula for double twist knots.
result Confirmed cyclotomic expansion conjecture for SU(N)-invariants.
We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.
Study on Einstein metrics on SU(3) Lie group, including new Lorentzian example.
problem Existence and properties of left-invariant pseudo-Riemannian Einstein metrics.
method Investigation of left-invariant metrics, use of isometry groups, analysis of curvatures.
result First Lorentzian homogeneous Einstein metric on SU(3).
There are two families of Donaldson invariants for the complex projective plane, corresponding to the SU(2)-gauge theory and the SO(3)-gauge theory with non-trivial Stiefel-Whitney class. In 1997 Moore and Witten conjectured that the regularized u-plane integral on the complex projective plane gives the generating func…
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…
Study on SU(2) group's Lorentzian problem, focusing on controllability and extremals.
problem Left-invariant Lorentzian problem on SU(2) group.
method Addressing complete controllability, existence of length maximizers, extremals.
result Results on controllability and extremals for the Lorentzian problem.
For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent submodules we establish simple conditions under which an invariant metric f-structure (f,…
In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into . We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existe…