scholarly journals Minimizing intersection points of curves under virtual homotopy

2020 ◽  
Vol 29 (03) ◽  
pp. 2050007
Author(s):  
Vladimir Chernov ◽  
David Freund ◽  
Rustam Sadykov
Keyword(s):  
Poisson Bracket ◽  
Homotopy Class ◽  
Finite Collection ◽  
Virtual Link ◽  
Closed Curves ◽  
Oriented Surface ◽  
Single Curve ◽  
Intersection Points

A flat virtual link is a finite collection of oriented closed curves [Formula: see text] on an oriented surface [Formula: see text] considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves [Formula: see text], we show that the minimal number of intersection points of curves in the virtual homotopy class of [Formula: see text] equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

scholarly journals Addendum: "(1, 2) and weak (1, 3) homotopies on knot projections"

2014 ◽  
Vol 23 (08) ◽  
pp. 1491001 ◽  
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Personal Communication ◽  
Finite Collection ◽  
Closed Curves ◽  
Oriented Surface ◽  
Reidemeister Moves ◽  
Closed Oriented Surface

After this paper was published, the following information about doodles was pointed out by Roger Fenn. A doodle was introduced by Fenn and Taylor [2], which is a finite collection of closed curves without triple intersections on a closed oriented surface considered up to the second flat Reidemeister moves with the condition (*) that each component has no self-intersections. Khovanov [4] introduced doodle groups, and for his process, he considered doodles under a more generalized setting (i.e. removing the condition (*) and permitting the first flat Reidemeister moves). He showed [4, Theorem 2.2], a result similar to our [3, Theorem 2.2(c)]. He also pointed out that [1, Corollary 2.8.9] gives a result similar to [4, Theorem 2.2]. The authors first noticed the above results by Fenn and Khovanov via personal communication with Fenn, and therefore, the authors would like to thank Roger Fenn for these references.

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

scholarly journals GEOMETRY OF THE HOMOLOGY CURVE COMPLEX

2012 ◽  
Vol 04 (03) ◽  
pp. 335-359 ◽  
Author(s):  
INGRID IRMER
Keyword(s):  
Homotopy Class ◽  
Intersection Number ◽  
Simple Algorithm ◽  
Curve Complex ◽  
Oriented Surface ◽  
Immersed Surfaces ◽  
Minimal Genus ◽  
Closed Oriented Surface

Suppose S is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, [Formula: see text], of S; a complex closely related to complexes studied by Bestvina–Bux–Margalit and Hatcher. A path in [Formula: see text] corresponds to a homotopy class of immersed surfaces in S × I. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in [Formula: see text], and for constructing minimal genus surfaces in S × I. It is proven that for g ≥ 3 the best possible bound on the distance between two vertices in [Formula: see text] depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For g ≥ 4 it is shown that [Formula: see text] is not δ-hyperbolic.

scholarly journals Venn Diagrams with Few Vertices

10.37236/1382 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Lower Bound ◽  
Planar Graph ◽  
Connected Region ◽  
Venn Diagram ◽  
Closed Curves ◽  
Venn Diagrams ◽  
Intersection Points ◽  
Few Vertices

An $n$-Venn diagram is a collection of $n$ finitely-intersecting simple closed curves in the plane, such that each of the $2^n$ sets $X_1 \cap X_2 \cap \cdots \cap X_n$, where each $X_i$ is the open interior or exterior of the $i$-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight $k$ is a $k$-region. A monotone Venn diagram with $n$ curves has the property that every $k$-region, where $0 < k < n$, is adjacent to at least one $(k-1)$-region and at least one $(k+1)$-region. Monotone diagrams are precisely those that can be drawn with all curves convex. An $n$-Venn diagram can be interpreted as a planar graph in which the intersection points of the curves are the vertices. For general Venn diagrams, the number of vertices is at least $ \lceil {2^n-2 \over n-1} \rceil$. Examples are given that demonstrate that this bound can be attained for $1 < n \le 7$. We show that each monotone Venn diagram has at least ${n \choose {\lfloor n/2 \rfloor}}$ vertices, and that this lower bound can be attained for all $n > 1$.

scholarly journals Kauffman–Jones polynomial of a curve on a surface

2017 ◽  
Vol 26 (11) ◽  
pp. 1750062
Author(s):  
Shinji Fukuhara ◽  
Yusuke Kuno
Keyword(s):  
Homotopy Class ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Laurent Polynomial ◽  
Computational Results ◽  
Oriented Surface ◽  
Variable Formula ◽  
Polynomial Formula

We introduce a Kauffman–Jones type polynomial [Formula: see text] for a curve [Formula: see text] on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial [Formula: see text] is a Laurent polynomial in one variable [Formula: see text] and is an invariant of the homotopy class of [Formula: see text]. As an application, we obtain an estimate in terms of the span of [Formula: see text] for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of [Formula: see text] and show some computational results on the span of [Formula: see text].

scholarly journals Complexity of virtual multistrings

2021 ◽  
pp. 2150066
Author(s):  
David Freund
Keyword(s):  
Closed Curves ◽  
Oriented Surface ◽  
Text String ◽  
Type 3 ◽  
Closed Oriented Surface

A virtual[Formula: see text]-string [Formula: see text] consists of a closed, oriented surface [Formula: see text] and a collection [Formula: see text] of [Formula: see text] oriented, closed curves immersed in [Formula: see text]. We consider virtual [Formula: see text]-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual 1-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [Formula: see text]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual 1-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [Formula: see text]-strings in general, but Gibson found a counterexample for 5-strings. We show that Kadokami’s statement holds for non-parallel [Formula: see text]-strings and exhibit a counterexample for general virtual 3-strings.

COMPLEX FENCHEL-NIELSEN COORDINATES FOR QUASI-FUCHSIAN STRUCTURES

1994 ◽  
Vol 05 (02) ◽  
pp. 239-251 ◽  
Author(s):  
SER PEOW TAN
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Oriented Surface ◽  
Hyperbolic Structures ◽  
Embedded Image ◽  
Totally Real ◽  
Real Subspace ◽  
Closed Oriented Surface

Let Fg be a closed oriented surface of genus g ≥ 2 and let [Formula: see text] be the space of marked quasi-fuchsian structures on Fg. Let [Formula: see text] be a set of non-intersecting, non-trivial simple closed curves on Fg that cuts Fg into pairs of pants components. In this note, we construct global complex coordinates for [Formula: see text] relative to [Formula: see text] giving an embedding of [Formula: see text] into [Formula: see text]. The totally real subspace of [Formula: see text] with respect to these coordinates is the Teichmüller Space [Formula: see text] of marked hyperbolic structures on Fg, the coordinates reduce to the usual Fenchel-Nielsen coordinates for [Formula: see text] relative to [Formula: see text]. Various properties of the embedded image are studied.

scholarly journals On the Richter–Thomassen Conjecture about Pairwise Intersecting Closed Curves

2016 ◽  
Vol 25 (6) ◽  
pp. 941-958
Author(s):  
JÁNOS PACH ◽  
NATAN RUBIN ◽  
GÁBOR TARDOS
Keyword(s):  
Real Function ◽  
Important Ingredient ◽  
Continuous Real Function ◽  
Closed Curves ◽  
Jordan Curves ◽  
Family Of Curves ◽  
The Family ◽  
Intersection Points ◽  
Pass Through

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1−o(1))n2.We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.)An important ingredient of our proofs is the following statement. Let S be a family of n open curves in ℝ2, so that each curve is the graph of a continuous real function defined on ℝ, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$.

Curves, Surfaces, and Hyperbolic Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Homotopy Class ◽  
Hyperbolic Geometry ◽  
Hyperbolic Plane ◽  
Closed Surface ◽  
Intersection Numbers ◽  
Hyperbolic Surfaces ◽  
Closed Curves

This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a useful tool since each homotopy class of simple closed curves has a unique geodesic representative. The chapter begins by recalling some basic results about surfaces and hyperbolic geometry, with particular emphasis on the boundary of the hyperbolic plane and hyperbolic surfaces. It then considers simple closed curves in a surface S, along with geodesics and intersection numbers. It also discusses the bigon criterion, homotopy versus isotopy for simple closed curves, and arcs. Finally, it describes the change of coordinates principle and three facts about homeomorphisms.

scholarly journals ROOTS OF DEHN TWISTS ABOUT MULTICURVES

2017 ◽  
Vol 60 (3) ◽  
pp. 555-583 ◽  
Author(s):  
KASHYAP RAJEEVSARATHY ◽  
PRAHLAD VAIDYANATHAN
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Mapping Class ◽  
Finite Collection ◽  
Closed Curves ◽  
Dehn Twists ◽  
Left Handed ◽  
The Individual ◽  
Necessary And Sufficient ◽  
Combinatorial Data

AbstractA multicurve${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist$t_{\mathcal{C}}$about${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$. In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

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