arXiv research

The paper modifies a basis for quantum topology and studies a weight system on trees.

The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…

Paper generalizes Kreweras triangle using universal sl_2 weight system.

Research disproves the extension of a weight system to a 4-invariant for graphs.

Constructs a 4-invariant for graphs at c = 3/8.

Formula for $\mathfrak{sl}_2$ weight system on complete bipartite graphs.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

We give a construction of Kirby weight systems associated to sl(2) and valued into the finite field Z/pZ. We show that it is possible to apply this sequence of weight systems on the universal invariant of framed link. We also show that the corresponding sequence admits a Fermat limit, which defines an asymptotic ration…

We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small o…

New stable homotopy types for knots and diagrams, aiding in their computation.

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

New invariant constructed using stable homotopy methods.

Develops method to construct Lie algebra weight system kernel using Vogel algebra.

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree …

Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.

New weight systems derived from a specific Lie algebra for knot invariants.

To recover a sparse signal from an underdetermined system, we often solve a constrained L1-norm minimization problem. In many cases, the signal sparsity and the recovery performance can be further improved by replacing the L1 norm with a "weighted" L1 norm. Without any prior information about nonzero elements of the si…

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

The paper classifies topological properties of specific geometric forms on manifolds.

Graphs and local systems count multiwebs.

The paper explores weight systems and their applications to graph and embedded graph invariants.

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

The paper defines a new homotopy type for colored links and proves stabilization behavior.

One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his cla…

SL2MF predicts synthetic lethality using logistic matrix factorization.

Categorifies weight-preserving maps in sl(2) representations.

We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle …

Paper generalizes representations of Lie algebroids to weighted Lie algebroids.

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

Most of the existing methods for sparse signal recovery assume a static system: the unknown signal is a finite-length vector for which a fixed set of linear measurements and a sparse representation basis are available and an L1-norm minimization program is solved for the reconstruction. However, the same representation…

Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…

Novel symmetry found in colored HOMFLY polynomials from superalgebras.

Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T^* R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2). The deformations of Pol(T^* R), which become trivial once…

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

Second part of proving linearization theorem for sl2(C).

Study Poisson cohomology and linearize Lie algebra structures.

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a defor…

Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.

New systems of linear PDEs discovered in 3D contact manifolds.

Computes Lie algebra structure constants using a graphical calculus.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

Proves generic independence and additivity of SL(2,C) Casson-Lin invariant.

We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of $U_q(sl_2)$. We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…

The paper develops a new theory for quantum link invariants using quandles and biquandles.

Quantum cluster algebra constructed from web skein relations on surfaces.

Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.

We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…