arXiv research

Unified approach to experimental design using interlacing polynomials.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeis…

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph b…

In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and…

New concept of regular separation for ODEs leads to improved Hardy field results.

We characterize those unions of embedded disjoint circles in the 2-sphere which can be the multiple point set of a generic immersion of the 2-sphere into 3-dimensional space in terms of the interlacement of the given circles. Our result is the one higher dimensional analogue of Rosenstiehl's characterization of words b…

Study eigenvector overlaps in large Gaussian matrices, simplifying for GOE.

Loops in surfaces and chord diagrams are studied with graph factorizations and grammars.

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $σ_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estima…

Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way …

DNA sequencing to identify genetic variants is becoming increasingly valuable in clinical settings. Assessment of variants in such sequencing data is commonly implemented through Bayesian heuristic algorithms. Machine learning has shown great promise in improving on these variant calls, but the input for these is still…

New framework estimates eigenvalues of kernel matrices without full matrix construction.

Node centrality is one of the most important and widely used concepts in the study of complex networks. Here, we extend the paradigm of node centrality in financial and economic networks to consider the changes of node "importance" produced not only by the variation of the topology of the system but also as a consequen…

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this doub…

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

This paper establishes certain existence and classification results for solutions to $SU(n)$ Toda systems with three singular sources at 0, 1, and $\infty$. First, we determine the necessary conditions for such an $SU(n)$ Toda system to be related to an $n$th order hypergeometric equation. Then, we construct solutions …

We propose a model in which a spliced vector bundle (with an arbitrary number of gauge structures in the splice) possesses a geometry which do not split. The model employs connection 1-forms with values in a space-product of Lie algebras, and therefore interlaces the various gauge structures in a non-trivial manner. Sp…

The present paper is a review of the current state of Graph-Link Theory (graph-links are also closely related to homotopy classes of looped interlacement graphs), dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. I…

Weaved helices form mechanically stable 3D structures.

Equation discovery methods enable modelers to combine domain-specific knowledge and system identification to construct models most suitable for a selected modeling task. The method described and evaluated in this paper can be used as a nonlinear system identification method for gray-box modeling. It consists of two int…

Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.

The paper examines circle graphs of Gauss diagrams and finds counterexamples to previous descriptions.

Paper proposes a robust framework for detecting multiple periodic components in time series.

This work explains how tempering improves Bayesian neural networks by reducing the impact of data augmentation.

New $F$-polynomial distinguishes knotoid diagrams not previously possible.

The paper studies polynomials and ideals from colored Jones polynomials for links.

Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.

This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.

Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.

The paper defines and classifies Cappell-Shaneson polynomials.

Developed algorithms to compute three polynomial invariants of veering triangulations.

Study links weaving knots with polynomial coefficients and lattice numbers.

Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.

Paper connects AJ conjecture and colored Jones polynomial potential function.

Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…

This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.

The taut polynomial equals a twisted Alexander polynomial.

Quantum polynomials are derived from a specific tribracket structure.

We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomial…

New polynomials detect non-rotatable knotoid shapes.

Innovates polynomial invariant for tribrackets.

Researchers extend Alexander polynomial to knotoids and linkoids.

Unified ADO and colored Jones polynomials for knots.

Generalized quandle polynomial used for stuquandles, stuck links, and RNA folding.

In this paper, we address the problem of synthesizing multi-parameter magnetic resonance imaging (mp-MRI) data, i.e. Apparent Diffusion Coefficients (ADC) and T2-weighted (T2w), containing clinically significant (CS) prostate cancer (PCa) via semi-supervised adversarial learning. Specifically, our synthesizer generates…

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, gen…

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author. The relation between this…