scholarly journals Distance and intersection number in the curve graph of a surface

2021 ◽  
Author(s):  
Joan S. Birman ◽  
Matthew J. Morse ◽  
Nancy C. Wrinkle
Keyword(s):  
Intersection Number ◽  
Curve Graph

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.

scholarly journals An algebraic formula for the intersection number of a polynomial immersion

2010 ◽  
Vol 214 (3) ◽  
pp. 269-280 ◽  
Author(s):  
Iwona Karolkiewicz ◽  
Aleksandra Nowel ◽  
Zbigniew Szafraniec
Keyword(s):  
Intersection Number ◽  
Algebraic Formula

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

2018 ◽  
Vol 61 (1) ◽  
pp. 195-230 ◽  
Author(s):  
JESÚS HERNÁNDEZ HERNÁNDEZ
Keyword(s):  
Topological Type ◽  
Orientable Surface ◽  
Finite Topological Type ◽  
Curve Graph ◽  
Finite Set ◽  
Rigid Set

AbstractFor an orientable surfaceSof finite topological type with genusg≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph$\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger,J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger,Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

scholarly journals Computing the Geometric Intersection Number of Curves

2019 ◽  
Vol 66 (6) ◽  
pp. 1-49
Author(s):  
Vincent Despré ◽  
Francis Lazarus
Keyword(s):  
Intersection Number

A remark on elementary contractions

1995 ◽  
Vol 118 (1) ◽  
pp. 183-188
Author(s):  
Qi Zhang
Keyword(s):  
Projective Variety ◽  
Intersection Number ◽  
Smooth Projective Variety ◽  
Complex Numbers ◽  
Canonical Bundle ◽  
Small Contraction ◽  
Dimension One ◽  
Extremal Ray

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

SOME NUMERICAL INVARIANTS OF FLAT VIRTUAL LINKS ARE ALWAYS REALIZED BY REDUCED DIAGRAMS

2006 ◽  
Vol 15 (03) ◽  
pp. 289-297 ◽  
Author(s):  
TERUHISA KADOKAMI
Keyword(s):  
Intersection Number ◽  
Finite Sequence ◽  
Closed Surface ◽  
Crossing Number ◽  
The Self ◽  
Virtual Link ◽  
Virtual Knot ◽  
Virtual Links ◽  
Geodesic Loop ◽  
Numerical Invariants

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

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.

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