Distances and Intersections of Curves

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.

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Pseudo-Anosov Theory

10.23943/princeton/9780691147949.003.0015 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Braid Groups ◽

Intersection Numbers ◽

Branched Covers ◽

Algebraic Integers ◽

Dehn Twists ◽

Mapping Classes ◽

Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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

manuscripta mathematica ◽

10.1007/s00229-019-01157-2 ◽

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.

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Small intersection numbers in the curve graph

Bulletin of the London Mathematical Society ◽

10.1112/blms/bdu057 ◽

2014 ◽

Vol 46 (5) ◽

pp. 989-1002 ◽

Cited By ~ 3

Author(s):

Tarik Aougab ◽

Samuel J. Taylor

Keyword(s):

Intersection Numbers ◽

Curve Graph

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SELF-INTERSECTION NUMBERS OF PATHS IN COMPACT SURFACES

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511008826 ◽

2011 ◽

Vol 20 (03) ◽

pp. 403-410 ◽

Cited By ~ 3

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.

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Intersection numbers in the curve graph with a uniform constant

Topology and its Applications ◽

10.1016/j.topol.2016.03.009 ◽

2016 ◽

Vol 204 ◽

pp. 157-167 ◽

Cited By ~ 3

Author(s):

Yohsuke Watanabe

Keyword(s):

Intersection Numbers ◽

Curve Graph

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

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511008887 ◽

2011 ◽

Vol 20 (03) ◽

pp. 469-496 ◽

Cited By ~ 2

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.

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Dehn Twists

10.23943/princeton/9780691147949.003.0004 ◽

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.

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