A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
New method analyzes knots and links using multiscale Gauss link integral.
We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map fr…
In this paper, we shall give an explicit Gauss diagram formula for the Kontsevich integral of links up to degree four. This practical formula enables us to actually compute the Kontsevich integral in a combinatorial way.
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integ…
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…
Formula connects curvature to volume in special geometric spaces.
Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.
The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking …
The Gauss self-linking integral of an unframed knot is not a knot invariant, but it can be turned into an invariant by adding a correction term which requires adding extra structure to the knot. We collect the different definitions/theorems/proofs concerning this correction term, most of which are well-known (at least …
Study on multiple linking numbers, extending Gauss diagram formulas.
A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. W…
We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body and its polar body . The Mahler conjecture asserts that the Mahler volume is minimiz…
We construct radial fundamental solutions for the differential form Laplacian on negatively curved symmetric spaces. At least one of these Green's functions also yields a Biot-Savart Opearator, i.e. a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential. Any Biot-Sav…
The paper defines new representations and groups related to virtual links.
Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previo…
In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surf…
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.
Defines Vassiliev complexity measures for open and closed curves in 3D space.
New formulas for spatial 2-bouquet graphs discovered.
Generalizes Gauss-Bonnet to metrics with logarithmic singularities.
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by coun…
New method calculates knot and link properties using state codes.
We introduce here explicit integral formulas for linking, twisting, writhing and helicity on the 3-sphere and in hyperbolic 3-space. These formulas, like their prototypes in Euclidean 3-space, are geometric rather than just topological, in the sense that their integrands are invariant under orientation-preserving isome…
We provide a proof and analyze the asymptotic behavior of a formula for the linking number of line segments.
Integrability of the (2+1)-dimensional Gauss-Codazzi-Mainardi equation is considered. It is shown that this equation is the particular cases of the Yang-Mills-Higgs-Bogomolny and self-dual Yang-Mills equations.
Paper introduces simplified formulas for Milnor's triple linking number.
Given a construction of smooth homotopy class invariants of smooth immersions . The particular case of is a sequence of non-zero integrals, where the term is the Gauss-Bonnet integral
Solves optimal stopping for Gauss-Markov bridges using time-space transformation.
Revisits the Gauss-Bonnet formula using double forms.
In this first of two papers, we develop a steady-state version of classical electrodynamics on the 3-sphere and in hyperbolic 3-space, including an explicit formula for the vector-valued Green's operator, an explicit formula of Biot-Savart type for the magnetic field, and a corresponding Ampere's Law contained in Maxwe…
Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/ge…
We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots (or knots with ordered Gauss code) and their product.
We investigate cobordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cobordant to the trivial one.
For a compact differentiable surface with boundary embedded in , we give simple proofs of the Gauss-Bonnet theorem, Poincaré-Hopf theorem, and several other integral formulas. We complete all of the proofs without using fundamental or differential forms.
Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This pap…
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.
We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …
A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting su…
The paper connects linking numbers to Biot-Savart kernels on compact manifolds.
In this article, we give the non-integrated defect relations for the Gauss map of a complete minimal surface with finite total curvature in This is a continuation of previous work of Ha-Trao [J. Math. Anal. Appl., \textbf{430} (2015), 76-84.], which we extend here to targets of higher dimension.
The classical integral localization formula for equivariantly closed forms (Theorem 7.11 in [BGV]) is well-known and requires the acting Lie group to be compact. It is restated here as Theorem 2. In this article we extend this result to NONcompact groups. The main result is Theorem 20. Then, using this generalization, …
The aim of this paper is to give a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a d…
Harmonic and minimal great circle fibrations have special Gauss maps.
Sharp lower bound found for integral varifolds' mean curvature.
We consider the energy and bienergy functionals as variational problems on the set of Riemannian metrics and present a study of the biharmonic stress-energy tensor. This approach is then applied to characterise weak conformality of the Gauss map of a submanifold. Finally, working at the level of functionals, we recover…