arXiv research

The paper extends surface link coloring theory to triplane diagrams and knots.

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for…

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…

We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized Kontsevitch integral (which is consequently not the universal invariant of finite …

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 …

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classi…

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which we call bikei. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, …

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander module…

Paper develops a method to learn causal networks with non-invertible functions.

We define invariants of unoriented knots and links by enhancing the integral kei counting invariant Phi_X^Z (K) for a finite kei X using representations of the kei algebra, Z_K[X], a quotient of the quandle algebra Z[X] defined by Andruskiewitsch and Grana. We give an example that demonstrates that the enhanced invaria…

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 …

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.

Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \mathbb{Z} [x_1,x_2,...,x_1^{-1},x_2^{-1},...])$, where the number of variabl…

The paper solves a problem in constructing a bicategory of algebra bundles.

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian.…

Learning domain-invariant representations has become a popular approach to unsupervised domain adaptation and is often justified by invoking a particular suite of theoretical results. We argue that there are two significant flaws in such arguments. First, the results in question hold only for a fixed representation and…

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

We prove the existence of a degree 7 Vassiliev invariant of long (or string) two-component links which is not preserved under the simultaneous change of orientation of both components. The non-invertibility of this invariant can be detected by the standard weight system with values in the tensor square of the universal…

Study nonparametric factor analysis with arbitrary noise.

We prove an adiabatic decomposition formula of the zeta-determinant of the Laplace type operator with respect to Dirichlet boundary condition. We allow the non-invertible tangential operator. As a result, our adiabatic decomposition formula involves the scattering matrix over the manifold with cylindrical end. We also …

Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.

Extended symmetries and anomalies in compactified 6d SCFTs on various internal manifolds.

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $σ$-connections on finitely generated projective modules. This ma…

Neural ODEs and i-ResNet are recently proposed methods for enforcing invertibility of residual neural models. Having a generic technique for constructing invertible models can open new avenues for advances in learning systems, but so far the question of whether Neural ODEs and i-ResNets can model any continuous inverti…

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…

While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…

Unified framework for learning with indirect supervision signals.

Unified framework for disentangled representations using mechanistic independence.

New framework improves classification accuracy using Pillai's trace and ULDA.

SurVAE Flows combine VAEs and flows using surjective transformations.

Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.

New methods for parameter estimation in mechanistic models using data-consistent inversion.

AIKAE enhances IKAE for long-term time series forecasting.

Theoretical framework for target propagation shows differences from backpropagation.

The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints in the form of distance comparisons like "item $i$ is closer to item $j$ than item $k$". Ordinal constraints like this often come from human judgments. To account for errors and variation in jud…

Study on estimating invertible functions with minimax analysis.

The paper explores how invertibility affects the complexity of encoder models in VAEs.

Polynomially parametrize interesting knotted surfaces.

New 2-knots found with same knot group but different quandles.

New knot quandles distinguish ribbon knots with isomorphic groups.

Proved colored HOMFLY-PT polynomials for specific knots.

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transver…

Study concordance of alternating torus knots to L-space knots.

This paper studies how knots combine using Alexander Polynomials.

The paper classifies a special family of knots in lens spaces using knot Floer homology.