arXiv research

Algorithm calculates Jones polynomial from Goeritz matrix.

The paper defines and classifies Cappell-Shaneson polynomials.

Paper extends Cohen's method to compute Jones polynomial for certain braid subfamilies.

Link signature limit depends on linking matrix under specific polynomial condition.

A new method computes link invariants from diagrams.

We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional…

Graphical notation simplifies complex polynomial constraints in linear models.

Invariants defined for braid systems under Hurwitz equivalence.

A simple multivariable version of the reduced Burau matrix is constructed for any braid. It is shown how the multivariable Alexander polynomial for the closure of the braid can be found directly from this matrix.

Harer-Zagier formulas generalized to knot matrix models.

New algorithm optimizes matrix reordering for noisy disordered matrices.

We analyze kernel matrices in polynomial high-dimensional settings and explain double descent in KRR.

Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.

Develops polynomial diffusion models for multi-factor commodity futures dynamics.

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably…

New method proves Jones Polynomial's connect sum property.

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynom…

Simplified Khovanov polynomials for bipartite links.

Develops a Gaussian model to compute the Alexander polynomial of knots.

We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial $L_α(G)=D-\sum_{r=1}^dα_rD(D^{-1}A)^r$ where $A$ is the adjacency matrix of a weighted, undirected graph, $D$ is the diagonal matrix of weighted degrees, and $α=(α_1...α_d)$ are nonn…

A new method calculates HOMFLY-PT polynomials for bipartite links.

Deep tensor factorization benefits from implicit regularization with polynomial growth.

Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.

The Kashaev conjecture is proven for classical signatures and Alexander polynomials of links.

The paper infers multiple graphs from stationary signals on them.

We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.

We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. …

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

This paper studies HOMFLY polynomials of specific and infinite classes of knots.

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte'…

Simplified KR polynomial for bipartite links reduces to tensor products of vector spaces.

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

We solve principal component regression (PCR), up to a multiplicative accuracy $1+γ$, by reducing the problem to $\tilde{O}(γ^{-1})$ black-box calls of ridge regression. Therefore, our algorithm does not require any explicit construction of the top principal components, and is suitable for large-scale PCR instances. In…

The paper defines and computes a knot complement invariant for simple links.

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its …

Starting from the free field realization of Kac-Moody Lie algebra, we define a generalized Yang-Yang function. Then for the Lie algebra of type $A_{n}$, we derive braiding and fusion matrix by braiding the thimble from the generalized Yang-Yang function. One can construct a knots invariant $H(K)$ from the braiding and …

We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.

We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Che…

We develop a method to factorize symmetric sparse Boolean matrices efficiently.

The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We u…

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…

The paper explores how low-degree polynomials can detect shuffled linear regression models.

Classifies matrices in the quaternionic hyperbolic unitary group.

The paper models financial correlation matrices using permutation invariant Gaussian models and predicts market anomalies.

Covariance pooling is a feature pooling method with good classification accuracy. Because covariance features consist of second-order statistics, the scale of the feature elements are varied. Therefore, normalizing covariance features using a matrix square root affects the performance improvement. When pooling methods …

This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm…

This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and …