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.
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.
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.
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.
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. 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. 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.
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…
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.
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. For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…
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].
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…
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.
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.
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…
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.
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,…
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.
We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] on manifolds to graphs: For a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dime…
There has been a large amount of interest, both in the past and particularly recently, into the power of different families of universal approximators, e.g. ReLU networks, polynomials, rational functions. However, current research has focused almost exclusively on understanding this problem in a worst-case setting, e.g…
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.
Investigates polynomial time algorithms for computing Khovanov homology of braids.
problem Computing Khovanov homology for general braids is intractable.
method Examines polynomial time algorithms for 3-braids and a variation of the scanning algorithm for more general braids.
result Shows that for 3-braids, Khovanov homology can be computed in polynomial time, while for more general braids, it can be computed in polynomial time for bounded homological degrees.
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.
We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.
We give a refined upper bound for the hyperbolic volume of an alternating link in terms of the first three and the last three coefficients of its colored Jones polynomial.
The study optimizes polynomial regression for learning under Gaussian distributions.
problem Agnostic learning of Boolean and real-valued functions under Gaussian distributions.
method LP duality and polynomial degree analysis for L1-regression. result Optimal SQ lower bounds for various function classes.
In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is also discussed.