相关论文: Roots in 3-manifold topology
Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…
In this paper, we give a new proof of the foundational result, due to S. Cutkosky, on the existence of a monomialisation of a morphism from a 3-fold to a surface. Our proof brings to the fore the notion of log-Fitting ideals, and requires…
We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus…
Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we…
In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any…
We prove that the Davenport-Mahler bound holds for arbitrary graphs with vertices on the set of roots of a given univariate polynomial with complex coefficients.
We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and…
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have…
Let $N=2n^2-1$ or $N=n^2+n-1$, for any $n\ge 2$. Let $M=\frac{N-1}{2}$. We construct families of prime knots with Jones polynomials $(-1)^M\sum_{k=-M}^{M} (-1)^kt^k$. Such polynomials have Mahler measure equal to $1$. If $N$ is prime, these…
The set $\mathit{RT}(M)$ of values of the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be both finite and infinite. We prove that $\mathit{RT}(M)$ is a finite set if $M$ is the splice of two certain knots in the…
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove…
A graph is one-ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex $v$ {\em dominates} a ray in the…
Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.
We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The…
This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph…
It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in…
The aim of the work is to prove the following main theorem. Theorem. Let M3 be a three-dimensional, connected, simple-connected, closed, compact, smooth manifold. Tnen the manifold M3 is diffeomorphic to the three-dimensional sphere.
New obstructions for embedding one compact oriented 3-manifold in another are given. A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein-theoretic generalizations for 2n-tangles…
Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the…
We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper…