The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.
problem Investigating the roots of Alexander polynomials of random positive 3-strand braids.
method Experimental data analysis, conjectures refinement, and proof of results using tools like the signature function of links and Lyapunov exponent of the Burau representation.
result Generically, at least 69% of the roots of Alexander polynomials are on the unit circle, with a large root-free region near the origin.
Random polynomial dynamical systems often have negative Lyapunov exponents.
problem Understanding the behavior of random polynomial dynamical systems.
method Investigation of i.i.d. random complex dynamical systems generated by probability measures.
result For a generic system, the Lyapunov exponent of almost every sequence of maps is negative for most initial values.
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective…
Polynomial-time algorithm matches correlated random graphs with non-vanishing correlation.
problem Matching correlated random graphs with non-vanishing edge correlation.
method Iterative algorithm for polynomial-time recovery of latent matching.
result Algorithm succeeds in recovering latent matching as long as edge correlation is non-vanishing.
Jones polynomials have infinitely many roots of unity as zeros.
problem Finding roots of unity as zeros of Jones polynomials.
method Constructing families of prime knots with specific Jones polynomials.
result Infinitely many roots of unity are zeros of some Jones polynomials.
Study shows dense roots of Yamada polynomial for certain graphs.
problem Understanding the roots of Yamada polynomials for spatial graphs.
method Construction and analysis of Yamada polynomial for spatial graphs.
result Found an infinite family of graphs with dense roots of Yamada polynomials.
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
Polynomials' roots count tied to surface umbilics.
problem Relating roots of polynomials to umbilics on surfaces.
method Constructing a convex surface from a polynomial, determining umbilic index, and applying Hamburger's bound.
result Bounding the number of roots inside the unit circle for polynomials with self-inversive second derivatives.
Polyhedron volume is a multiple root of its polynomial.
problem Understanding the volume of flexible polyhedra.
method Proof of volume polynomial properties.
result Volume of flexible polyhedra is a multiple root of its polynomial.
Classifies trees with strictly unimodal q-polynomials.
problem Classifying rooted trees with strictly unimodal q-polynomials.
method Classification based on plucking polynomials and criteria for trapezoidal shapes.
result Generalizes results on strict unimodality of q-binomial coefficients.
The paper defines when two trees have identical plucking polynomials.
problem Identifying trees with the same plucking polynomial.
method Provided a sufficient and necessary condition for trees with identical plucking polynomials.
result A sequence of non-negative integers can be realized as a rooted tree.
Study on unimodality of plucking polynomial with delay function.
problem Exploring unimodality of plucking polynomial with delay function.
method Presented a formula for the plucking polynomial of hedgehog rooted trees and explored unimodality with specific delay functions.
result Found interesting examples and speculations on unimodality of plucking polynomials with delay functions.
Paper finds periodic patterns in colored Jones polynomial values.
problem Understanding periodicity in colored Jones polynomial values.
method Investigated polynomial values at specific roots of unity and -1.
result Periodic patterns in polynomial values at certain substitutions.
The interior polynomial of a bipartite graph's hypergraph equals its Ehrhart polynomial of its root polytope.
problem Understanding the relationship between the interior polynomial of a bipartite graph and its root polytope.
method Proving equivalence between the interior polynomial of a bipartite graph's hypergraph and the Ehrhart polynomial of its root polytope.
result The interior polynomials of a bipartite graph and its transpose agree.
Categorifies colored Jones polynomial at roots of unity.
problem Categorification of colored Jones polynomial.
method Differential on triply-graded homology, compatible with p-differential structure.
result Categorification of the colored Jones polynomial at a root of unity.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
Study polynomials' root clustering via symmetric products, unifying stability approaches.
problem Stability of polynomials with roots in specific regions.
method Interpreting roots and coefficients as symmetric product morphism, analyzing topology up to homeomorphism.
result Description of strata topology and adjacency, explaining classical stability problems.
For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.
This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; how…
We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial Lα(G)=D−∑r=1dαrD(D−1A)r where A is the adjacency matrix of a weighted, undirected graph, D is the diagonal matrix of weighted degrees, and α=(α1...αd) are nonn…
Bounds on knot polynomials for Lie superalgebras of type I.
problem Determining genus bounds for knot polynomials colored by Lie superalgebra representations.
method Proved bounds on the t-degree of knot polynomials, relating it to the number of odd roots and the genus of the knot. result Proved bounds on knot polynomials for Lie superalgebras of type I, showing equality for certain knots.
Continuity of polynomial roots shown for varying coefficients.
problem Continuity of polynomial roots under varying coefficients.
method Uniform bounds and Sobolev space analysis.
result Solution map is continuous for Cd coefficients. Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Asymptotic expansion formula for colored Jones polynomial using twist knots.
result Obtained asymptotic expansion formulas for twist knots at specified roots of unity.
Study finds root vertex in large networks with high probability.
problem Finding the root vertex in large growing networks.
method Constructs confidence sets for the root vertex in various random network models.
result Confidence sets of size independent of the number of vertices contain the root vertex with high probability.
Study on quantum invariants of twist knots at specific roots of unity.
problem Asymptotic expansions of quantum invariants for twist knots.
method Saddle point method applied to colored Jones polynomial.
result Asymptotic expansion formula for twist knots at given root of unity.
Investigates the topology of polynomial singularities, improving bounds and comparing to random cases.
problem Understanding the topological complexity and structure of polynomial singularities.
method Analyzes polynomial maps and their singularities, using semialgebraic sets and Betti numbers.
result Improved upper bound on topological complexity and comparison to random cases.
Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.
problem Positivity conjecture for Roger--Yang skein algebras.
method Used explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases.
result Polynomials form a lower bound in the sense of [Lê18] and [LTY21].
Study shows link polynomial evaluations from Heegaard Floer theory.
problem Link polynomial evaluations from Heegaard Floer theory.
method Definition of Euler characteristic for fractionally-graded complexes based on roots of unity.
result Equality of Alexander polynomial evaluations and sl(n) polynomial evaluations at certain roots of unity. We introduce certain polynomials, so-called H.Weyl and H.Minkowski polynomials, which have a geometric origin. The location of roots of these polynomials is studied.
Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).
problem Understanding the convergence rate of polynomial networks to Gaussian processes.
method Examined one-hidden-layer neural networks with random weights, focusing on polynomial activations and their convergence rate in the 2-Wasserstein metric.
result The rate of convergence for polynomial networks to Gaussian processes is $O(n^{- rac{1}{2}})$.
Study on Jones polynomials and their roots in the unit circle and complex plane.
problem Understanding the roots of Jones polynomials for knots and links.
method Analyzing solutions of the equation JK(t)=1 for double-twist knots and links. result The set of solutions to JKn(t)=1 is dense in the unit circle and complex plane. Compact bilinear pooling approximates covariance features for faster training.
problem Efficiently approximating covariance features for faster training.
method Compact bilinear pooling extended to polynomial approximations of covariance features.
result The proposed method achieves comparable accuracy with fewer dimensions.
The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot group…
We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the h-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of p…
We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This result is applied to prove that when M is closed, it dominates at most finitely ma…
We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruen…
We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.
Researchers map the fundamental group of polynomial strata to a braid group.
problem Understanding the fundamental group of polynomial strata.
method Analyzing the logarithmic derivative of polynomials to determine the map to a braid group.
result The map from the fundamental group of a stratum to a braid group is characterized by the geometry of the translation surface structure.
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for appro…
The Burau representation fails to be faithful at roots of unity for n ≥ 3.
problem The faithfulness of the Burau representation at roots of unity.
method Analyzing the braid element σiσi+1σi and its powers. result The Burau representation is unfaithful at any primitive root of unity, excepting the first three.
New link homologies categorify Jones polynomial at odd prime powers.
problem Categorify Jones polynomial at odd prime powers.
method Specialize Cautis differential to positive integers.
result Non-isomorphic link homologies for odd primes.
A sequence fn(q) is q-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in q and qn. Our main theorems state that q-holonomicity is preserved under twisting, i.e., replacing q by ωq where ω is a complex root of unity, and under the substitution q→qα where $α…
Majority bit estimation in noisy random recursive DAGs.
problem Estimating the majority bit in a noisy random recursive DAG.
method Majority rule among nodes, with bit flipping and noisy channel.
result Identification of the threshold for p at which majority rule yields errors. Minimal surfaces from simple polynomials solve a geometric problem.
problem Constructing minimal surfaces using Traizet's method.
method Using polynomials that satisfy a hypergeometric differential equation.
result Simple minimal surfaces are described by these polynomials.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
Study spider evaluations and web group representations using algebraic topology.
problem Evaluate and represent the topology of web groups using algebraic methods.
method Analyze the fundamental groups of planar webs, associate irreducible components and graded subalgebras, and use Poincaré polynomials.
result Spider evaluations correspond to symmetrized Poincaré polynomials of associated graded subalgebras.
From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a generalization of the property H_r = (H_1)^r for special polynomials at q=1. We conjecture …