arXiv research

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.

Study on knots, genera, and algebraic concordance groups.

To construct mirror symmetric Landau-Ginzburg models, P.Berglund, T.Hübsch and M.Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold Euler characteristics…

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibilit…

Study on 2-bridge knots, proving equivariant concordance order is infinite.

Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine …

The study connects knot complements to 3d theories via half-index calculations.

The paper defines higher invariants for groups of polynomial growth and proves their convergence.

A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links. Furthermore, it …

P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construct…

New formulas connect knot invariants with theta functions.

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

New examples show Sasaki manifolds without extremal metrics.

It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular $2x2$ matrices. In this work, we propose to generalize this result by considering the representations…

Paper develops polynomial approximations for complex probability densities.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…

A new polynomial invariant for strongly involutive links.

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…

We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in …

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by attempting to solve a non-convex optimization problem involving the generator. In this …

For every odd integer $c\ge 21$, we raise an example of a prime component-preservingly amphicheiral link with the minimal crossing number $c$. The link has two components, and consists of an unknot and a knot which is $(-)$-amphicheiral with odd minimal crossing number. We call the latter knot a {\it Stoimenow knot}. W…

We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat β$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the b…

This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate r…

Study on estimating invertible functions with minimax analysis.

PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.

The paper provides non-asymptotic Edgeworth expansions for neural network outputs.

Local invertibility of higher order tensor transforms on compact manifolds.

Study of strongly invertible Legendrian links in contact 3-space.

Method estimates observation functions in state-space models without supervision.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

Let G be a Lie group with Lie algebra $ \Cal G: = T_εG$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter…

Dirac operator invertibility proven for specific manifolds.

Table of symmetric diagrams for knots up to 10 crossings.

ISR creates analytical relationships from data via invertible maps.

Local invertibility of ray transforms on convex manifolds.

Let $φ: S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $φ$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $φ$ has no critical points at all). We introduce …

This work tackles exploding inverses in INNs, revealing and mitigating their numerical non-invertibility.

CF-INNs can approximate any invertible function, resolving a long-standing problem.

This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.

Develops equivariant grid homology for strongly invertible knots.

New findings on knot genera using advanced techniques.

Defines knot signature invariant using G-signature theorem.

New invariants derived from Seifert graphs help distinguish alternating links.

Paper shows invertibility of tensor X-ray transform on certain manifolds.