New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
Polynomial neural networks explore thresholds for maximum expressiveness.
problem Understanding the limits of polynomial neural networks' expressiveness.
method Introducing activation degree threshold to measure network expressiveness and proving its existence and upper bounds.
result Polynomial neural networks with equi-width architectures achieve the maximum expressiveness.
Sharp upper bound for quasi polynomial degree of manifold configuration spaces.
problem Determining the exact degree of quasi-polynomial homology groups of configuration spaces.
method Analyzing extremal homology groups of unordered configuration spaces of manifolds.
result The upper bound for the degree of quasi-polynomials is sharp for every manifold.
Low-degree method fails to predict robust subspace recovery problem.
problem Predicting computational tractability of robust subspace recovery problem.
method Low-degree polynomial framework, anti-concentration properties.
result Low-degree method fails to predict computational tractability of robust subspace recovery problem even up to high degree.
New findings on computational limits for estimating hidden structures.
problem Estimating hidden structures in noisy data.
method Use of low-degree polynomials as a restricted model of computation.
result Established low-degree hardness of recovery problems for easy detection problems.
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
Alexander polynomial degree correlates with knot defect, proving conjecture for defect zero.
problem Characterizing knot polynomials and their defects.
method Analyzing differential expansions and degree in q±2 of Alexander polynomials. result Proved Alexander polynomial degree correlates with knot defect, especially for defect zero.
Homology groups of spaces of nonsingular polynomial embeddings R1→Rn of degrees ≤4 are calculated. A general algebraic technique of such calculations for spaces of polynomial knots of arbitrary degrees is described.
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.
We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how the Jones slopes and the linear term behave under cabling. As an application we ve…
This note gives a proof that the A-polynomial of any nontrivial knot in S3 has nontrivial M-degree.
Study disproves conjecture about low-degree polynomials in hypothesis testing.
problem Conjecture about limitations of polynomial-time algorithms in hypothesis testing.
method Used counterexamples to refute the conjecture and modified the conjecture to rule out the counterexample.
result Disproved conjecture about limitations of low-degree polynomials in hypothesis testing.
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses d2-adic expansion instead of 4-adic for higher degree polynomials. result Provides a solution that depends on the d2-adic expansion of the power of the mapping class element. The paper explores how low-degree polynomials can detect shuffled linear regression models.
problem Detecting multivariate shuffled linear regression models from independent Gaussian random matrices.
method Investigates the effectiveness of low-degree polynomial algorithms for distinguishing the model from independent Gaussian random matrices.
result Establishes a phase transition phenomenon in the performance of low-degree polynomial algorithms for distinguishing the model.
Enhances psyquandle invariants for singular and pseudoknots.
problem Counting invariants for singular knots and pseudoknots.
method Uses quivers to extend in-degree polynomial invariants.
result Obtains biquandle coloring quivers and in-degree polynomial invariants.
We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp…
New work shows FP potential monotonicity equals low-degree polynomial estimators limits.
problem Establishing a precise mathematical relationship between statistical physics and polynomial estimators limits.
method Analyzing Gaussian additive models (GAMs) to show FP potential monotonicity equals low-degree polynomial estimators limits.
result For a broad family of Gaussian additive models, the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential.
In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov-Polyak-Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree ≤n if and only if an invariant is a polynomial of degree ≤n on every twist lattice of the right form. The main resul…
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
New bound on Jones polynomial for specific positive links.
problem Finding bounds on the Jones polynomial for positive links.
method Using previous results on positive fibered links, we found a new bound for a specific family of positive links.
result We provided a bound on the maximum degree of the Jones polynomial for positive links with a specific coefficient.
Two new polynomial invariants for long virtual knots.
problem Defining new polynomial invariants for long virtual knots.
method Introducing V1(K;t) and V2(K;t), establishing properties, and showing realizability. result First derivatives of V1(K;t) and V2(K;t) at t=1 define finite type invariants of degree three. Enhanced invariant for linkoids using quivers.
problem Counting invariants for linkoids.
method Use of quivers to generalize in-degree polynomial invariant.
result Introduced in-degree quiver polynomial matrix as a new invariant.
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
New knots with specific properties have identical polynomial values.
problem Identifying knots with matching polynomial values after braiding.
method Constructing infinitely many hyperbolic knots and analyzing their braided satellites.
result Mutually distinct hyperbolic knots have identical HOMFLY polynomial values up to given z-degrees. New bounds on HOMFLY polynomial for homogeneous links.
problem Bounding the minimum v-degree of HOMFLY polynomial for homogeneous links. method Proved a slice version of Cromwell's inequality and a related conjecture.
result New bounds on the minimum v-degree of HOMFLY polynomial for homogeneous links. New upper bound on Jones polynomial for fibered positive links.
problem Classifying positive and non-positive knots of crossing number ≤ 12.
method Proved a new upper bound on the maximum degree of Jones polynomial for fibered positive knots.
result Maximum degree of Jones polynomial for fibered positive knots is at most four times the minimum degree.
In-degree quiver polynomials for surface-links computed.
problem Computing in-degree quiver polynomials for surface-links.
method Defined using a quandle and set of endomorphisms, computed for surface-links with ch-index up to 10.
result Example computations for surface-links with ch-index up to 10.
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
Study robustness of polynomial neural networks using algebraic geometry.
problem Certify robustness radius of polynomial neural networks.
method Metric algebraic geometry, Euclidean distance degree, symbolic elimination, homotopy-continuation methods.
result Found decision boundaries with lower ED degree than generic cubic hypersurfaces.
The colored HOMLFY polynomial is an important knot invariant depending on two variables a and q. We give bounds on the degree in both a and q generalizing Morton's bounds \cite{Mo86} for the ordinary HOMFLY polynomial. Our bounds suggest that the degree detects certain incompressible surfaces in the knot comple…
We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^{(m,n)}(b,0)=d^m/da^m d^n/dx^n P_K(a,x)|(a, x) = (b, 0) of the HOMFLY polynomial of a knot K at (b,0) is a Vassiliev invariant …
New method uses almost orthonormal bases to prove low-degree lower bounds in complex statistical models.
problem Proving statistical-computational gaps in high-dimensional models with planted structures.
method Constructing an almost orthonormal polynomial basis under the planted distribution.
result Established new low-degree lower bounds for various complex models.
New evidence shows computational barriers in graphon estimation using low-degree polynomials.
problem Estimating graphons efficiently and accurately.
method Low-degree polynomials to analyze computational limits.
result Low-degree polynomial estimators cannot significantly outperform USVT in graphon estimation.
Gradient Descent with Projection learns low-degree polynomials efficiently.
problem Learning low-degree spherical polynomials with neural networks.
method Over-parameterized two-layer neural network with Gradient Descent with Projection.
result Achieves nearly minimax optimal sample complexity and risk bound.
Algorithm creates polynomials for knotted surfaces, with bounds on degree.
problem Creating polynomials for knotted surfaces with constraints.
method Algorithm constructs polynomials based on loop braids and surface braids.
result Upper bounds on the degree of polynomials for knotted surfaces.
New bounds on Khovanov homology for positive links families.
problem Bounding the maximum non-vanishing quantum degree of Khovanov homology.
method Diagram-independent upper bounds constructed using Khovanov homology.
result Constructs upper bounds on Khovanov homology, providing positivity obstructions.
Jones slopes detect figure eight knot, and characterize alternating knots.
problem Detecting knots using Jones polynomials.
method Strong slope conjecture and colored Jones polynomials.
result Jones slopes detect figure eight knot and characterize alternating knots.
We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For…
We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot K satisfies the Slope Conjecture then a (p,q)-cable of K satisfies the conjecture, provided that p/q is not a Jon…
Non-negative L1-approximating polynomials for Gaussian distributions are proven for certain classes of sets.
problem Existence of non-negative L1-approximating polynomials for Gaussian distributions. method Proving the existence of degree-k non-negative polynomials that approximate indicator functions of sets with Gaussian surface area in L1-norm. result Proves the existence of non-negative L1-approximating polynomials for certain classes of sets with Gaussian surface area. We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a (2,q)-torus knot.
We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational tangles.
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
Study the expressivity and training complexity of polynomial neural networks.
problem Understanding the expressivity and training complexity of polynomial neural networks.
method Use algebraic geometry to describe neuromanifolds and neurovarieties, analyzing their dimension and learning degree.
result Characterized the dimension and learning degree of neuromanifolds, providing geometric and complexity measures.
In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We a…
We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.