scholarly journals ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES

2020 ◽  
Vol 8 ◽  
Author(s):  
VINCENT DELECROIX ◽  
ÉLISE GOUJARD ◽  
PETER ZOGRAF ◽  
ANTON ZORICH
Keyword(s):  
Moduli Spaces ◽  
Simple Closed Curve ◽  
Vertical Cylinder ◽  
Topological Type ◽  
Closed Curve ◽  
Theoretical Physics ◽  
Asymptotic Formulas ◽  
Quadratic Differentials ◽  
Closed Curves ◽  
Polymer Folding

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

scholarly journals Monotone homotopies and contracting discs on Riemannian surfaces

2018 ◽  
Vol 10 (02) ◽  
pp. 323-354 ◽  
Author(s):  
Gregory R. Chambers ◽  
Regina Rotman
Keyword(s):  
Finite Volume ◽  
Minimal Surfaces ◽  
Analogous Result ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Riemannian Surface ◽  
Closed Curves ◽  
Minimal Hypersurfaces ◽  
Riemannian Surfaces ◽  
Complete Manifolds

A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].

Taming Wild Simple Closed Curves with Monotone Maps

1972 ◽  
Vol 24 (5) ◽  
pp. 768-788
Author(s):  
W. S. Boyd ◽  
A. H. Wright
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Curves ◽  
Monotone Map ◽  
Monotone Maps

Hempel [6, Theorem 2] proved that if S is a tame 2-sphere in E3 and f is a map of E3 onto itself such that f|S is a homeomorphism and f(E3 - S) = E3- f(S), then f(S) is tame. Boyd [4] has shown that the converse is false; in fact, if S is any 2-sphere in E3, then there is a monotone map f of E3 onto itself such that f |S is a homeomorphism, f(E3 — S) = E3 — f(S), and f(S) is tame.It is the purpose of this paper to prove that the corresponding converse for simple closed curves in E3 is also false. We show in Theorem 4 that if J is any simple closed curve in a closed orientable 3-manifold M3, then there is a monotone map f : M3 → S3 such that f |J is a homeomorphism, f(J) is tame and unknotted, and f(M3 - J) = S3 - f(J).In Theorem 1 of § 2, we construct a cube-with-handles neighbourhood of a simple closed curve in an orientable 3-manifold.

scholarly journals Normal closures of powers of Dehn twists in mapping class groups

1992 ◽  
Vol 34 (3) ◽  
pp. 314-317 ◽  
Author(s):  
Stephen P. Humphries
Keyword(s):  
Class Group ◽  
Simple Closed Curve ◽  
Isotopy Class ◽  
Dehn Twist ◽  
Closed Curve ◽  
Mapping Class ◽  
Class Groups ◽  
Closed Curves ◽  
Dehn Twists ◽  
Definition Of

LetF = F(g, n)be an oriented surface of genusg≥1withn<2boundary components and letM(F)be its mapping class group. ThenM(F)is generated by Dehn twists about a finite number of non-bounding simple closed curves inF([6, 5]). See [1] for the definition of a Dehn twist. Letebe a non-bounding simple closed curve inFand letEdenote the isotopy class of the Dehn twist aboute. LetNbe the normal closure ofE2inM(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:Theorem 1.The subgroup N is of finite index in M(F).

Continuous, Slope-Preserving Maps of Simple Closed Curves

1980 ◽  
Vol 32 (5) ◽  
pp. 1102-1113
Author(s):  
Tibor Bisztriczky ◽  
Ivan Rival
Keyword(s):  
Unit Circle ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Curves ◽  
Continuous Map ◽  
Continuous Maps

How many of the continuous maps of a simple closed curve to itself are slope-preserving? For the unit circle S1 with centre (0, 0), a continuous map σ of S1 to S1 is slope-preserving if and only if σ is the identity map [σ(x, y) = (x, y)] or σ is the antipodal map [σ(x, y) = (–x, –y)]. Besides the identity map, more general simple closed curves can also possess an “antipodal” map (cf. Figure 1).

Limits in 𝒫ℳℱ of Teichmüller geodesics

2019 ◽  
Vol 2019 (747) ◽  
pp. 1-44 ◽  
Author(s):  
Jon Chaika ◽  
Howard Masur ◽  
Michael Wolf
Keyword(s):  
Line Segment ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Quadratic Differentials ◽  
Limit Set ◽  
Unique Point ◽  
Time Intervals ◽  
Limit Sets ◽  
Bounded Distance ◽  
Uniquely Ergodic

AbstractIn this paper we consider the limit set in Thurston’s compactification{\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}}of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.

scholarly journals INTERPOLATING STRING FIELD THEORIES

1992 ◽  
Vol 07 (12) ◽  
pp. 1079-1090 ◽  
Author(s):  
BARTON ZWIEBACH
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Open String ◽  
Field Theories ◽  
Quadratic Differentials ◽  
Closed Curves ◽  
First Order ◽  
Area Problem ◽  
Closed Theory ◽  
Minimal Area

A minimal area problem imposing different length conditions on open and closed curves is shown to define a one-parameter family of covariant open-closed quantum string field theories. These interpolate from a recently proposed factorizable open-closed theory up to an extended version of Witten’s open string field theory capable of incorporating on shell closed strings. The string diagrams of the latter define a new decomposition of the moduli spaces of Riemann surfaces with punctures and boundaries based on quadratic differentials with both first order and second order poles.

scholarly journals Arrangements of Pseudocircles: Triangles and Drawings

2020 ◽  
Vol 65 (1) ◽  
pp. 261-278
Author(s):  
Stefan Felsner ◽  
Manfred Scheucher
Keyword(s):  
Lower Bound ◽  
Final Section ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Recursive Construction ◽  
Closed Curves ◽  
Generation Algorithm ◽  
Combinatorial Data

AbstractA pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $$p_3$$ p 3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least $$2n-4$$ 2 n - 4 . We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with $$p_3({\mathscr {A}})/n \rightarrow 16/11 = 1.\overline{45}$$ p 3 ( A ) / n → 16 / 11 = 1 . 45 ¯ . We expect that the lower bound $$p_3({\mathscr {A}}) \ge 4n/3$$ p 3 ( A ) ≥ 4 n / 3 is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of n pairwise intersecting circles have at least $$2n-4$$ 2 n - 4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of $$p_3 \ge 2n/3$$ p 3 ≥ 2 n / 3 , and conjecture that $$p_3 \ge n-1$$ p 3 ≥ n - 1 . Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that $$p_3 \le \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) +O(n)$$ p 3 ≤ 4 3 n 2 + O ( n ) . This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with $$p_3 = \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ p 3 = 4 3 n 2 . The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.

scholarly journals RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

2021 ◽  
pp. 1-6
Author(s):  
Maxim Kazarian
Keyword(s):  
Moduli Spaces ◽  
Recursion Relation ◽  
Quadratic Differentials ◽  
Hodge Integrals

Abstract We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

scholarly journals Representing Homology Classes on Surfaces

1976 ◽  
Vol 19 (3) ◽  
pp. 373-374 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Unit Circle ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

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