Related papers: Finite Order q-Invariants of Immersions of Surface…

Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential…

Given a germ of a smooth plane curve $(\{f(x,y)=0\},0)\subset (\mathbb K^2,0), \mathbb K=\mathbb R, \mathbb C$, with an isolated singularity, we define two invariants $I_f$ and $V_f\in \mathbb N\cup\{\infty\}$ which count the number of…

An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new…

The purpose of this paper is to interpret polynomial invariants of strongly invertible links in terms of Khovanov homology theory. To a divide, that is a proper generic immersion of a finite number of copies of the unit interval and circles…

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections…

Let $f:\mathbb{C}\rightarrow \mathbb{R}^3$ be complete Willmore immersion with $\int_{\Sigma}|A_f|^2<+\infty$. We will show that if $f$ is the limit of an embedded surface sequence, then $f$ is a plane. As an application, we prove that if…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

We investigate constraints on embeddings of a non-orientable surface in a $4$-manifold with the homology of $M \times I$, where $M$ is a rational homology $3$-sphere. The constraints take the form of inequalities involving the genus and…

In [LMO] a 3-manifold invariant $\Omega(M)$ is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant $\Omega$ takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here we show that…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…

In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such…

A $Q$-manifold $M$ is a supermanifold endowed with an odd vector field $Q$ squaring to zero. The Lie derivative $L_Q$ along $Q$ makes the algebra of smooth tensor fields on $M$ into a differential algebra. In this paper, we define and study…

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants…

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Besides, we introduce the finite Chen type…

We show that every morphism from a degree 5 hypersurface in 4-dimensional projective space to a nonsingular degree 3 hypersurface in 4-dimensional projective space is necessarily constant. In the process, we also classify morphisms from the…

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped…

Let $M$ be a sphere with handles and holes, $f:M\to\mathbb R^3$ an embedding, and $H_1=H_1(M;\mathbb Z)$. We study a simple isotopy invariant of $f$, the Seifert bilinear form $L(f):H_1\times H_1\to\mathbb Z$. Let $\cap:H_1\times…

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…

Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies.…