arXiv research

New formula recovers degree of colored Jones polynomials for pretzel knots.

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to pres…

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…

A formula for the difference of Vassiliev invariants of degree k+1 of two knots all of whose Vassiliev invariants of degree k agree is proven. The proof uses K. Habiro's C-moves and his theorem which relates them to Vassiliev invariants.

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…

We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to …

Formula counts rational curves with a specific singular point in projective space.

Formula connects linking coefficients to Kontsevich integral coefficients.

Formula derived for Gromov-Witten invariants of smooth curves.

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.

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…

We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mariño-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases o…

We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as …

We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…

We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these formulas always represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to c…

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…

We give an explicit formula for the projectively invariant quantization map between the space of symbols of degree three and the space of third-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on a manifold.

Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface ha…

Efficient algorithm for identifying causal effects in linear models.

In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …

We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.

We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form $\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$ where $N$ is the set of com…

We extend Kac-Rice formula to compute expected intersections of random submanifolds.

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space M{g,n} in the sense of orbifolds where it can be written in terms of Mumford-M…

Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain the geometric meaning of this series as degrees of maps of some grand configurat…

Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres …

Abstract: Generalizes modular forms to family case and finds new anomaly cancellation formulas.

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomi…

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection …

Study on symplectic semi-characteristic using cohomology and vector fields.

Derives a new formula for measuring risk aversion in markets.

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…

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reduc…

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2), (d_1,...,d_m), for several values of g and h. We obtain explicit formulae of arithmetic…

We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a sy…

Study robustness of polynomial neural networks using algebraic geometry.

A new formula detects differences between counterexamples and standard embeddings of circles.

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behavi…

The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Mat…

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

A new measure quantifies how risk-averse different risk measures are.

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…

Two new rational formulae for normal implied volatility are presented.

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our for…