SELF-INTERSECTION NUMBERS OF PATHS IN COMPACT SURFACES

2011 ◽  
Vol 20 (03) ◽  
pp. 403-410 ◽  
Author(s):  
LORENA ARMAS-SANABRIA ◽  
FRANCISCO GONZÁLEZ-ACUÑA ◽  
JESÚS RODRÍGUEZ-VIORATO
Keyword(s):  
Free Group ◽  
Intersection Number ◽  
Compact Surface ◽  
Simple Path ◽  
Intersection Numbers ◽  
Minimal Self ◽  
Compact Surfaces

In this paper, we give an algorithm to calculate the minimal self-intersection number of paths in a compact surface with boundary representing a given element of the free group F(x1, x2, …, xn). In particular, this algorithm says whether or not a word in x1, x2, …, xn is representable by a simple path. Our algorithm is simpler than similar algorithms given previously. In the case of a disk with n holes the problem is equivalent to the problem of deciding which relators can appear in an Artin n-presentation.

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

scholarly journals Distances and Intersections of Curves

2018 ◽  
Vol 2020 (23) ◽  
pp. 9674-9693
Author(s):  
Yohsuke Watanabe
Keyword(s):  
Intersection Number ◽  
Intersection Numbers ◽  
Dehn Twists ◽  
Curve Graph ◽  
Mapping Classes

Abstract We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we study intersection numbers of curves contained in geodesics in the curve graph. Furthermore, we generalize a well-known result on intersection number growth of curves under iteration of Dehn twists and multitwists for all kinds of pure mapping classes.

THE JONES POLYNOMIAL AND THE INTERSECTION NUMBERS OF TWISTED CYCLES ASSOCIATED WITH A SELBERG TYPE INTEGRAL

2011 ◽  
Vol 20 (03) ◽  
pp. 469-496 ◽  
Author(s):  
KATSUHISA MIMACHI
Keyword(s):  
Special Functions ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Homology Theory ◽  
Intersection Numbers ◽  
Type Integral ◽  
Jones Polynomials ◽  
Twisted Homology ◽  
Hypergeometric Type ◽  
Definition Of

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

scholarly journals A Note on a Multiplicity Result for the Mean Field Equation on Compact Surfaces

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Aleks Jevnikar
Keyword(s):  
Field Equation ◽  
Mean Field ◽  
Compact Surface ◽  
Multiplicity Result ◽  
The Mean ◽  
Compact Surfaces ◽  
Mean Field Equation

AbstractWe are concerned with the class of equations with exponential nonlinearitieson a compact surface Σ, which describes the mean field equation of equilibrium turbulence with arbitrarily signed vortices. Here,

scholarly journals A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-sphere

2021 ◽  
Author(s):  
Shosaku Matsuzaki
Keyword(s):  
Compact Surface ◽  
Reidemeister Moves ◽  
Trivalent Graphs ◽  
Embedded Surfaces ◽  
Compact Surfaces

We give a presentation for a non-split compact surface embedded in the 3-sphere [Formula: see text] by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces represented by the diagrams are ambient isotopic in [Formula: see text].

scholarly journals Growth of intersection numbers for free group automorphisms

2010 ◽  
Vol 3 (2) ◽  
pp. 280-310 ◽  
Author(s):  
Jason Behrstock ◽  
Mladen Bestvina ◽  
Matt Clay
Keyword(s):  
Free Group ◽  
Intersection Numbers ◽  
Group Automorphisms ◽  
Free Group Automorphisms

Dehn Twists

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Free Group ◽  
Mapping Class Group ◽  
Infinite Order ◽  
Class Group ◽  
Intersection Number ◽  
Mapping Class ◽  
Algebraic Operation ◽  
Closed Curves ◽  
Dehn Twists ◽  
Rank 2

This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

scholarly journals Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature

1987 ◽  
Vol 7 (1) ◽  
pp. 49-72 ◽  
Author(s):  
J. Feldman ◽  
D. Ornstein
Keyword(s):  
Invariant Measure ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Surface ◽  
Maximal Entropy ◽  
Unit Tangent Bundle ◽  
Compact Surfaces ◽  
Measure Of Maximal Entropy

AbstractLet g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.

scholarly journals A generalization of Turaev’s virtual string cobracket and self-intersections of virtual strings

2016 ◽  
Vol 19 (04) ◽  
pp. 1650053 ◽  
Author(s):  
Patricia Cahn
Keyword(s):  
Homotopy Class ◽  
Intersection Number ◽  
Large Set ◽  
Minimal Self ◽  
Free Homotopy Class ◽  
Minimum Number ◽  
Number Formula ◽  
Matrix Invariant ◽  
Intersection Points ◽  
Virtual Strings

Previously we defined an operation [Formula: see text] that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that [Formula: see text] gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that [Formula: see text] gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings [Formula: see text] such that [Formula: see text] gives a formula for the minimal self-intersection number [Formula: see text]. Finally, we compare the bound given by [Formula: see text] to a bound given by Turaev’s based matrix invariant [Formula: see text], and construct an example that shows the bound on the minimal self-intersection number given by [Formula: see text] is sometimes stronger than the bound [Formula: see text].

scholarly journals Linearization of topologically Anosov homeomorphisms of non compact surfaces of genus zero and finite type

2021 ◽  
pp. 1-11
Author(s):  
Gonzalo Cousillas ◽  
Jorge Groisman ◽  
Juliana Xavier
Keyword(s):  
Finite Type ◽  
Compact Surface ◽  
Genus Zero ◽  
Compact Surfaces

We study the dynamics of {\it topologically Anosov} homeomorphisms of non-compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f\colon S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \mathbb{R}^2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in \cite{cgx}.

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