Study algebraic relations of Vassiliev invariants for families of knots.
problem Understanding algebraic structure of Vassiliev invariants for knot families.
method Analyzing algebraic relations and generating sets of Vassiliev invariants in 3D Chern-Simons theory.
result For 1-parametric knot families, Vassiliev invariants are finitely generated. For more parameters, there can be an infinite number of generators.
Knot invariants from XC-structures on Sweedler algebra are trivially determined.
problem Defining and characterizing knot invariants from XC-structures.
method Examining XC-structures on the Sweedler algebra and their relation to knot invariants.
result Knot invariants from XC-structures on Sweedler algebra are completely determined by the framing of the knot.
Refines a tangle invariant using XC-algebras.
problem Building a refined tangle invariant using XC-algebras.
method Constructing a canonical strict monoidal functor that refines the Kerler-Kauffman-Radford invariant.
result Preserves the braiding, twist, and open trace.
New invariant for singular links via bt-algebra.
problem Invariants for singular links.
method Representations of singular braid monoid into two parameter bt-algebra.
result More powerful than previous invariants.
The paper describes metrics on left Leibniz algebras, linking them to quadratic Lie algebras.
problem Understanding metrics on left Leibniz algebras and their connections to quadratic Lie algebras.
method Analyzing left multiplications, right multiplications, and bilinear forms on left Leibniz algebras.
result Left Leibniz algebras with associative metrics can be derived from their underlying quadratic Lie algebras.
Study algebraic invariants from lightning self-attention models.
problem Understanding polynomial coefficients of self-attention mechanisms.
method Identify algebraic invariants using polynomial coefficients and coordinate geometry.
result Found linear and nonlinear families of algebraic invariants.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
Paper connects two invariants of 3D manifolds using Hopf algebras.
problem Establishing a relation between two invariants of 3D manifolds.
method Using spherical Hopf algebras and their Drinfeld doubles, the paper connects the chromatic spherical invariant and the Hennings-Kauffman-Radford invariant.
result The chromatic spherical invariant is equal to the Hennings-Kauffman-Radford invariant for a specific type of Hopf algebra.
New examples of Lie algebras with ad-invariant metrics found.
problem Finding ad-invariant metrics on nonnice nilpotent Lie algebras.
method Introducing single extension method to construct Lie algebras with ad-invariant metrics.
result Explicit examples of nonnice nilpotent Lie algebras with ad-invariant metrics for dimensions > 10 and steps > 2.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
Vogel's construction links knot invariants to Lie algebras, revealing new insights.
problem Can all finite type knot invariants be derived from Lie algebras?
method Parameterized expansion coefficients with three parameters and constructed a polynomial to vanish for all simple Lie algebras.
result Vogel's construction implies an alternative axiomatization of simple Lie algebras.
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.
Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Study of coloured invariants of torus knots using W algebras.
problem Understanding coloured invariants of torus knots T(p,p′). method Representation theory of principal affine W algebras and asymptotic weight multiplicities. result Limits of renormalized invariants are equal to characters of W algebra modules. New method detects black hole horizons using Lie algebra invariants.
problem Detecting black hole horizons in spacetime.
method Scalar relative differential invariants with Lie algebra structure.
result General relative differential invariant vanishes on black hole horizons.
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra …
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf G-algebras and coloring Kirby diagrams. result Invariants defined for 4-manifolds and connections.
This paper updates knot invariants using Hopf algebras and categorifies their structure.
problem Defining and understanding quantum invariants of knots and three-manifolds.
method Abstract description of categorical structures involving Hopf algebras and their centers.
result The Hopf algebraic center of a knot's image is central to the invariant.
We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma--Hecke algebras. The isomorphism relates affine Yokonuma--Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma--Hecke algebras, and in turn, to produce …
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…
New Hopf algebras help classify 4D shapes.
problem Classifying 4D shapes up to deformations.
method Developed non-factorizable ribbon Hopf algebras.
result Some derived invariants are boundary-dependent.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
Modified Hennings invariant defined using quantum groups and integrals.
problem Defining a modified Hennings invariant using quantum groups.
method Topological ribbon Hopf algebra, discrete Fourier transforms, symmetrized graded integral, modified trace.
result Modified graded Hennings invariant defined and extended to empty manifolds.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the inva…
Invariant structures link to algebraic curves with specific properties.
problem Linking invariant hypercomplex structures to algebraic curves.
method Mapping invariant structures to algebraic curves with specific properties.
result Invariant hypercomplex structures correspond to algebraic curves with a flat projection and antiholomorphic involution.
A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category T of handlebody-tangles and present it by generators and relations. The result tells us that every functor on T that gives rise to inv…
We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-W…
The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf a…
New examples show algebraically slice knots with specific genus bounds.
problem Understanding slice genus of algebraic knots and their mirrors.
method Genus bound from Casson-Gordon invariants and cabling formula.
result Examples of algebraically slice knots with specific genus bounds.
New algebraic structure helps distinguish braids.
problem Distinguishing braids using mathematical invariants.
method Defined pointed racks and used them to create braiding invariants.
result New invariants can distinguish braids not previously possible.
New weight systems derived from a specific Lie algebra for knot invariants.
problem Constructing universal weight systems for knot invariants.
method Using a minimal Z22-graded Lie algebra to create weight systems. result Weight system derived from A1ε shows hybrid properties of sl(2) and gl(1∣1). We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to dias…
Study shows certain knots can't be sliced using 2-fold branched covers.
problem Determining which algebraically slice knots are actually slice.
method Used d invariants of 2-fold branched covers to show nonsliceness.
result Shows nonsliceness of a set of algebraically slice knots.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real…
New method finds invariants of Lie algebras, especially for semi-direct sums.
problem Finding invariants of semi-direct sums of Lie algebras.
method Equivalent problem of solving total differential equations instead of PDEs.
result All invariant functions of L are Casimir operators. In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions us…