Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
New lower bound on virtual crossing number using writhe polynomial.
problem Calculating the virtual crossing number of knots and links.
method New interpretation of the writhe polynomial to refine the lower bound.
result Refined lower bound on virtual crossing number.
Upper bound on Jones polynomials density modulo primes.
problem Density of Jones polynomials modulo prime numbers.
method Derived an upper bound on Jones polynomials density within a large degree range.
result Upper bound on Jones polynomials density modulo primes.
The paper improves bounds on the complexity of computing link polynomials.
problem Computing link polynomials by the skein relation is complex.
method Proved new upper and lower bounds on skein tree depth.
result New bounds on skein tree depth are stronger than previous ones.
Formula for colored Links-Gould polynomial with genus bounds.
problem Calculating polynomial for knots colored with specific representations.
method Cabling formula and genus bounds for the Links-Gould polynomial.
result Genus bounds and specialization to Alexander polynomial for colored Links-Gould polynomial.
Alexander polynomial degree bounds twice a knot's topological slice genus.
problem Determining the topological slice genus of knots.
method Using Freedman's disc theorem and Alexander polynomial properties.
result The degree of the Alexander polynomial is an upper bound for twice the topological slice genus.
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.
Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
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…
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.
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.
Optimal bounds found for ancient caloric functions on manifolds.
problem Bounding the dimension of ancient caloric functions on manifolds with polynomial volume growth.
method Analyzing polynomial growth and using Yau's conjecture for harmonic functions.
result Sharp bound for the dimension of ancient caloric functions on spaces where Yau's conjecture holds.
New bounds on surfaces in link complements are polynomial, not exponential.
problem Bounding the number of surfaces in link complements.
method Showed polynomial bound on genus g surfaces in complement of a prime alternating link.
result Number of genus g surfaces is bounded by a polynomial in n.
New bounds for knot complexity based on Jones polynomial coefficients.
problem Finding bounds for the crosscap number of knots and links.
method Using coefficients from the Jones polynomial, we derive two-sided bounds for Conway sums of strongly alternating tangles.
result Neither linear bound generalizes for all knots and links.
The study finds polynomial upper bounds for singularities in Einstein-scalar field system.
problem Understanding the strength of singularities in gravitational collapse.
method Analyzing geometric quantities, focusing on the Kretschmann scalar.
result Polynomial blow-up upper bounds O(1/rN) for the Kretschmann scalar, improving previous bounds. Homology handles with trivial Alexander polynomial bound a 3D sphere.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for Z-homology 3-spheres. result A distinguished homology handle with trivial Alexander polynomial bounds a homology S1imesD3. New knot polynomials yield simple results modulo primes.
problem Understanding knot polynomials modulo primes.
method Constructing knots with specific properties.
result All polynomials modulo p with bounded a-span are realizable by knots with bounded braid index. New bounds on Jones polynomial positivity for specific links.
problem Determining when Jones polynomial of positive links is non-negative.
method Developed new bounds and used them to obstruct positivity for infinitely many almost-positive diagrams.
result Infinitely many knots are classified as almost-positive.
New lower bound for knot genus using Links-Gould invariant.
problem Finding a tighter lower bound for knot genus.
method Representation theory of Uqgl(2∣1) to prove degree of Links-Gould polynomial bounds Seifert genus. result The Links-Gould polynomial provides a new lower bound on knot genus, detecting specific knots like Kinoshita-Terasaka and Conway.
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.
This paper investigates symmetric ribbon numbers of low-complexity knots.
problem Determining the minimum number of ribbon singularities in symmetric ribbon disks for knots with up to 12 crossings.
method Systematic investigation using knot polynomials and determinants.
result Novel lower bounds for symmetric ribbon numbers of knots with up to 12 crossings.
The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.
problem Improving bounds on skein tree depth and delta-crossing numbers for knots and links.
method Theoretical and computational analysis of skein trees and knot invariants.
result New upper and lower bounds on skein tree depth and delta-crossing numbers are derived.
If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily altern…
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. Polynomial bound on Reidemeister moves for each link type.
problem Recognizing whether a given link diagram represents a specific link type.
method Showed existence of a polynomial pK such that any two diagrams of a link type differ by at most pK(c1)+pK(c2) Reidemeister moves. result The problem of recognising a link type is in NP and can be completed in exponential time.
Explicit polynomial bound found for subgroup Dehn function.
problem Finding explicit bounds on Dehn functions of subgroups of hyperbolic groups.
method Constructing a specific example of a non-hyperbolic subgroup and analyzing its Dehn function.
result Explicit polynomial upper bound n96 on the Dehn function of a non-hyperbolic subgroup. New bounds on Seifert surfaces for alternating links are found.
problem Finding the number of Seifert surfaces of fixed genus for alternating links.
method Explicitly given polynomial bound for genus-g Seifert surfaces of fixed Euler characteristic.
result The number of genus-g Seifert surfaces is bounded by a polynomial in the number of crossings.
The study bounds positive bases of skein algebras using Chebyshev polynomials.
problem Finding positive bases in skein algebras of surfaces.
method Using Chebyshev polynomials to establish bounds.
result Normalized Chebyshev polynomials of type one give the only positive basis for the closed torus.
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
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].
New bounds for learning polynomial surrogates with L∞ guarantees.
problem Learning polynomial surrogates for bounded binary functions with L∞ error guarantees. method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as nd+1 for degree d polynomials and ns2 for sparse polynomials. 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 bounds for score matching in polynomial exponential families.
problem Understanding the sample complexity of score matching for polynomial exponential families.
method Non-asymptotic sample complexity analysis for score matching.
result First finite sample bounds for score matching in polynomial exponential families.
The study provides polynomial bounds for essential surfaces in various 3-manifolds.
problem Bounding the number of isotopy classes of embedded essential surfaces in 3-manifolds.
method Restricting to alternating link complements in 3-sphere, then extending results to other classes of cusped 3-manifolds.
result Explicit polynomial bounds on all embedded essential surfaces in 3-manifolds.
New knots bound rational homology balls, using Alexander polynomials.
problem Finding sliceness obstructions for knots.
method Computing twisted Alexander polynomials and simplifying their calculation.
result New non-slice knots with rational homology ball bounds.
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 analysis shows neural networks and low-degree polynomials perform well on sparse latent structure problems.
problem Understanding the performance of neural networks and polynomial approximators on real-world sparse latent structure problems.
method Analysis of neural networks and polynomial kernels of bounded degree on a simple, natural inference problem with sparse latent structure.
result Almost-tight bounds on the performance of neural networks and low-degree polynomials for the problem, showing qualitative differences from worst-case settings.
We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which implies the classical modulo 2 Murasugi congruence for knots. We also give sharp bou…
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…
The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.
problem Estimating dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
method Proved upper bounds for dimensions and existence of sections using polynomial growth.
result Upper bounds for dimensions and existence of holomorphic sections with polynomial growth on Kähler Ricci shrinkers.
We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar met…
The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result,…
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.
Develops methods to calculate global index of real polynomials.
problem Calculating the global index of real polynomials.
method Two methods: via atypical fibres and Milnor arcs clusters.
result Derives upper bounds for the global index, refining Durfee's degree-based bound.
New knot polynomials reveal patterns and mutations.
problem Understanding the structure and behavior of knot polynomials.
method Using a specific multivariable polynomial invariant derived from Nichols algebras.
result Emerging patterns in knot polynomials, including genus bounds and unexpected mutations.
Paper conjectures Links-Gould invariant generalizes Alexander polynomial.
problem Classifying knots and links using the Links-Gould invariant.
method Analyzing classical properties of the Links-Gould invariant.
result Evidence suggests Links-Gould invariant provides lower bounds for genus and fiberedness criteria.
Study ancient caloric functions on graphs, extending a theorem from manifolds.
problem Bounding the dimension of ancient caloric functions on graphs.
method Extending Colding and Minicozzi's theorem to graphs.
result Dimension of ancient caloric functions is bounded by growth degree and graph dimension.
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.