arXiv research

We use four dimensional techniques to derive general bounds on the $τ$ invariant of a satellite knot in $S^{3}$.

Survey on knot theory's impact on four-dimensional topology.

Authors show that genus defects of Hopf arborescent links are decidable.

The study computes invariants of satellite knots using bordered Floer homology.

Study knot invariants to answer questions about slice genus and clasp numbers.

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap …

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes …

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer…

New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordi…

Explains how knots relate to 4D shapes.

Machine learning finds knots that bound ribbon disks.

The paper characterizes and contrasts knots with high 4D clasp numbers.

New examples show clasp numbers can be zero yet four-genus can be arbitrarily large.

New surface observables yield 2-knot invariants in nonabelian theories.

Study inequalities between knot invariants and compute new bounds.

The study explores how surface diffeomorphisms of knots relate to their topological properties.

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solut…

It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a four-dimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a "Nahm pole" away from the knot with a further singularity along the…

We explain how the usual algebras of Feynman diagrams behave under the grope degree introduced in "Grope cobordism of classical knots." We show that the Kontsevich integral rationally classifies grope cobordisms of knots in 3-space when the ``class'' is used to organize gropes. This implies that the grope cobordism equ…

New weight systems derived from a specific Lie algebra for knot invariants.

For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus $n$-link. We compute economical presentations of knot groups for torus links using t…

Study finds all 4D neutral manifolds.

New types of Ricci solitons found in 4D Lorentzian geometry.

The study examines four-dimensional gradient Ricci solitons and their properties.

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.

In this note we prove that any four-dimensional half conformally flat gradient steady Ricci soliton must be either Bryant's soliton or Ricci flat. We also classify four-dimensional half conformally flat gradient shrinking Ricci solitons with bounded curvature.

Study finds all 4D Lie groups with harmonic curvature.

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbo…

Deep neural networks predict knot invariants across dimensions with high accuracy.

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

Investigates a new four-dimensional energy related to Willmore energy.

Study classifies 4D Ricci solitons with specific curvature conditions.

Unified framework designs LK structures using integer twists on non-manifold meshes.

We completely classify the algebraic Ricci solitons of four-dimensional pseudo-Riemannian generalized symmetric spaces.

Study on a specific obstruction in four-dimensional geometry.

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are…

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. T…

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a …

In this paper, we investigate geometric properties of some curvature tensors of a four-dimensional Walker manifold. Some characterization theorems are also obtained.

In the first of these two lectures, I describe a gauge theory approach to understanding quantum knot invariants as Laurent polynomials in a complex variable q. The two main steps are to reinterpret three-dimensional Chern-Simons gauge theory in four dimensional terms and then to apply electric-magnetic duality. The var…

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

Study on smoothness of 4D Willmore-type hypersurfaces.

Rigidity for 4D Willmore submanifolds with boundary.

The paper is devoted to the investigation of four-dimensional Kahler manifolds admitting non-affine H-projective mappings. We find all such manifolds which are non-Einstein. In the paper also Kahler manifolds admitting infinitesimal H-projective transformations are determined. It is proved that the class of Kahler mani…

Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.