A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.
Unified 3D R-matrices from quantum cluster algebra.
This paper describes a method to obtain state model parameters for an infinite series of Links-Gould link invariants LG^{m,n}, based on quantum R matrices associated with the (\dot{0}_m | \dotα_n) representations of the quantum superalgebras U_q[gl(m|n)]. Explicit details of the state models for the cases n=1 and m=1,2…
In this paper, we reconstruct Kuperberg's web space. We introduce a new web (a trivalent diagram) and new relations between Kuperberg's web diagrams and the new diagram. Using the webs, we define crossing formulas corresponding to R-matrices associated to some irreducible representations and calculate…
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equa…
The purpose of this paper is to establish a connection between various subjects such as dynamical r-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures developed in dg-ga/9508013 and dg-ga/9611001. In particular, we give a new method of classifying dynamical r-mat…
We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the geometric interpretation of their centers. The gauge invariance of the link invariant …
We review the Reshetikhin-Turaev approach to construction of non-compact knot invariants involving R-matrices associated with infinite-dimensional representations, primarily those made from Faddeev's quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as thei…
In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and -matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of…
We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…
In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe only single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological p…
Direct proof of Alexander polynomial scaling for L-shaped representations.
Study geometric bases for A-polynomials in SU(3) using arcade formalism.
Quantizes Chern-Simons invariant for tangle exteriors.
Integrable dynamics explained via geometric maps and cluster algebras.
We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical -matrices and matched pairs induced by Poisson group actions
fkcompute calculates a knot invariant from a braid presentation.
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L in S X I, a thickened, compact, oriented surface S, such that the link complement (S X I) - L has no essential vertical cylinder.
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…
The paper generalizes knot invariants and their connections to quivers and ideals.
For an arbitrary identity L=R between compositions of maps L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit…
In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new fi…
Somewhat unexpectedly, the study of the family of twisted knots revealed a hidden structure behind exclusive Racah matrices , which control non-associativity of the representation product in a peculiar channel . These are simultaneously symmetric and orthogo…
Study extends knot polynomials to links, identifying them with known invariants.
We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular -matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs o…
This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and Poisson homogeneous spaces of Poisson Lie groups. We hope that now we present this re…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371 corresponds to the case m=2, and our generalizations sheds additional light on …
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
Classifies Lie bialgebras using Darboux families.
New method for calculating HOMFLY polynomials in symmetric representations.
Quantum ML promises faster data analysis but faces trainability challenges.
QGAA learns latent quantum states, reducing errors in quantum data generation.
Quantum machine learning uses quantum cross entropy to minimize loss, but measurement loss affects this process.
Quantum Earth Mover's distance improves stability and efficiency in quantum learning.
Quantum Gaussian processes enable scalable quantum learning.
Quantum machine learning models can approximate any continuous function.
Quantum autoencoders allow for reducing the amount of resources in a quantum computation by mapping the original Hilbert space onto a reduced space with the relevant information. Recently, it was proposed to employ approximate quantum adders to implement quantum autoencoders in quantum technologies. Here, we carry out …
Introduces Quantum Data Center for quantum era benefits.
We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. Quantum hard Lefschetz theorem is proved. …
VQAs use classical optimization to train quantum circuits, promising quantum advantage.
Quantum optimization aids in financial crash prediction and portfolio management.
Quantum machine learning tackles large datasets with randomized measurements.
Post-quantum cryptography needed for blockchain security.
Quantum trace map connects Teichmüller theory and quantum groups.
Quantum states can be learned efficiently using gentle measurements.
A new hybrid framework reduces quantum runtime and noise effects.