Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
We give conditions on a knot on which the Morton-Franks-Williams inequality is not sharp. As applications, we show infinitely many examples of knots where the inequality is not sharp and also prove (by giving examples) that the deficit of the inequality can be arbitrarily large.
We generalize the Morton-Franks-Williams inequality to the colored sl(N) link homology defined in arXiv:0907.0695, which gives infinitely many new bounds for the braid index and the self linking number. A key ingredient of our proof is a composition product for the general MOY graph polynomial, which gener…
We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…
Short note on braid index and quasipositivity of certain pretzel knots.
problem Calculating braid index and identifying quasipositive status for specific pretzel knots.
method Used Morton-Franks-Williams inequalities and Khovanov-Rozansky concordance homomorphisms.
result Determined braid index and identified quasipositivity for knots with even crossings in one strand.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fa…
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the …
We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid inde…
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a lin…
In Dunfield's catalog of the hyperbolic manifolds in the SnapPy census which are complements of L-space knots in S3, we determine that 22 have tunnel number 2 while the remaining all have tunnel number 1. Notably, these 22 manifolds contain 9 asymmetric L-space knot complements. Furthermore, using SnapPy a…
Let β be a braid on n strands, with exponent sum w. Let Δ be the Garside half-twist braid. We prove that the coefficient of vw−n+1 in the Homfly polynomial of the closure of β agrees with (−1)n−1 times the coefficient of vw+n2−1 in the Homfly polynomial of the closure of βΔ2. This coinciden…
In this chapter (Chapter V) we present several results which demonstrate a close connection and useful exchange of ideas between graph theory and knot theory. These disciplines were shown to be related from the time of Tait (if not Listing) but the great flow of ideas started only after Jones discoveries. The first dee…
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. New inequality on sphere generalizes circle inequality.
problem Generalizing circle inequality to sphere.
method Develops a new inequality on the sphere that incorporates mass center deviation.
result Improves Aubin's inequality and Onofri's inequality.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
The paper finds new inequalities for convex polygons.
problem Finding precise inequalities for convex polygons.
method Analytic isoperimetric inequalities based on Schur convex functions, followed by Bonnesen-style and inverse Bonnesen-style inequalities.
result Sharp discrete isoperimetric inequalities for planar convex polygons.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Proves inequalities on curved spaces with positive curvature.
problem Proving inequalities on manifolds with nonnegative Ricci curvature.
method Analyzes manifolds with nonnegative Ricci curvature and Euclidean volume growth.
result Proves Heisenberg-Pauli-Weyl, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Paper refines Talagrand inequality on Euclidean spaces.
problem Improving Talagrand inequality for Euclidean spaces.
method Symmetrization and alternative proof methods.
result Several refined functional inequalities derived.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.
problem Proving a Harnack inequality for positive solutions to heat equations on Finsler metric measure manifolds.
method Volume comparison theorem, weighted Poincaré inequality, local uniform Sobolev inequality, mean value inequalities.
result Derives a Harnack inequality for positive solutions to heat equations.
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.
Extends Riemannian geometry inequalities with sharper estimates.
problem Deriving new inequalities on Riemannian manifolds.
method Investigates advanced Hardy and Rellich-type inequalities on complete noncompact manifolds with weight functions.
result Provides sharper estimates conforming to the geometry and structure of the manifold.
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
The study establishes inequalities on path space for sub-Riemannian manifolds.
problem Understanding functional inequalities on path space for sub-Riemannian manifolds.
method Derivative and integration by parts formulae on path space with respect to a natural gradient operator, showing bounds of horizontal Ricci curvature.
result Established functional inequalities on path space analogous to Riemannian geometry.
Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.
problem Finding optimal systolic inequalities for manifolds with complex cohomology structures.
method Extends Gromov's inequality to manifolds with fundamental cohomology classes as cup products of 2-dimensional classes.
result Provides an optimal systolic inequality for a new class of manifolds.
Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.
problem Investigate functional and geometric inequalities on hyperbolic spaces and Riemannian manifolds.
method Employ symmetrization and semigroup approach based on sharp estimates for heat semigroup.
result Developed robust inequalities and methods relying on geometric and isoperimetric properties.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.
Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Extended a mathematical inequality by Andrews.
problem Mathematical inequality by Andrews.
method Proved extensions of Andrews inequality.
result Extended mathematical inequality by Andrews.
Study Hardy identities and inequalities on Cartan-Hadamard manifolds.
problem Existence and nonexistence of extremal functions in Hardy inequalities.
method Using the notion of a Bessel pair, we derive Hardy identities and inequalities.
result Established several Hardy type inequalities with improvements and understandings.