scholarly journals An algorithm to classify rational 3-tangles

2015 ◽  
Vol 24 (01) ◽  
pp. 1550004 ◽  
Author(s):  
B. Kwon
Keyword(s):  
Disjoint Union ◽  
Closed Curves ◽  
Curves On Surfaces

A 3-tangle T is the disjoint union of three properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from (B3, T) to (D2 × I, {x1, x2, x3} × I). Two rational 3-tangles T and T′ are isotopic if there is an orientation-preserving self-homeomorphism h : (B3, T) → (B3, T′) that is the identity map on the boundary. In this paper, we give an algorithm to check whether or not two rational 3-tangles are isotopic by using a modified version of Dehn's method for classifying simple closed curves on surfaces.

scholarly journals Packing Curves on Surfaces with Few Intersections

2017 ◽  
Vol 2019 (16) ◽  
pp. 5205-5217 ◽  
Author(s):  
Tarik Aougab ◽  
Ian Biringer ◽  
Jonah Gaster
Keyword(s):  
Sharp Estimate ◽  
Circle Packing ◽  
Hyperbolic Surface ◽  
Closed Curves ◽  
Topological Surface ◽  
Curves On Surfaces

Abstract Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as $|\chi(S)|^{k^{2}+k+1}$. In this article, we narrow Przytycki’s bounds, obtaining \[ \mathcal{N}_{k}(S) =O \left( \frac{ |\chi|^{3k}}{ ( \log |\chi| )^2 } \right)\!. \] In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a circle packing argument of Aougab and Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we use a different argument to show \[ \mathcal{N}_{k}(S) \leq O(n^{2k+2}). \] This may be improved when $k=2$, and we obtain the sharp estimate $\mathcal{N}_2(S)=\Theta(n^3)$.

scholarly journals Filling of closed surfaces

2018 ◽  
Vol 10 (04) ◽  
pp. 897-913 ◽  
Author(s):  
Bidyut Sanki
Keyword(s):  
Disjoint Union ◽  
Mapping Class ◽  
Constructive Proof ◽  
Closed Curves ◽  
Closed Surfaces ◽  
Minimal Filling ◽  
Minimum Number ◽  
Genus Formula ◽  
Positive Integers ◽  
Closed Oriented Surface

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.

scholarly journals AN ALGEBRAIC CHARACTERIZATION OF SIMPLE CLOSED CURVES ON SURFACES WITH BOUNDARY

2010 ◽  
Vol 02 (03) ◽  
pp. 395-417 ◽  
Author(s):  
MOIRA CHAS ◽  
FABIANA KRONGOLD
Keyword(s):  
Conjugacy Class ◽  
Fundamental Group ◽  
Intersection Number ◽  
Algebraic Characterization ◽  
Closed Curves ◽  
Curves On Surfaces

We prove that a conjugacy class in the fundamental group of a surface with boundary is represented by a power of a simple curve if and only if the Goldman bracket of two different powers of this class, one of them larger than two, is zero. The main theorem actually counts self-intersection number of a primitive class by counting the number of terms of the Goldman bracket of two distinct powers, one of them larger than two.

INTERSECTIONS OF CURVES ON SURFACES WITH DISK FAMILIES IN HANDLEBODIES

2006 ◽  
Vol 15 (05) ◽  
pp. 631-649 ◽  
Author(s):  
JOEL ZABLOW
Keyword(s):  
Coordinate System ◽  
Dehn Twist ◽  
Heegaard Splitting ◽  
Closed Curves ◽  
Dehn Twists ◽  
Sufficiency Condition ◽  
Strongly Irreducible ◽  
Curves On Surfaces ◽  
Counting Formula

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection [Formula: see text] as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

scholarly journals Pseudo-Anosov maps and simple closed curves on surfaces

2000 ◽  
Vol 128 (2) ◽  
pp. 321-326 ◽  
Author(s):  
SHICHENG WANG ◽  
YING-QING WU ◽  
QING ZHOU
Keyword(s):  
Closed Curves ◽  
Curves On Surfaces

scholarly journals Dehn’s algorithm for simple diagrams

2015 ◽  
Vol 24 (14) ◽  
pp. 1550074
Author(s):  
Charles Frohman ◽  
Joanna Kania-Bartoszynska
Keyword(s):  
Conjugacy Class ◽  
Disjoint Union ◽  
Closed Surface ◽  
Intersection Numbers ◽  
Closed Curves ◽  
Class A ◽  
Positive Genus ◽  
Monotonically Decrease ◽  
Standard Presentation ◽  
Closed Oriented Surface

Dehn gave an algorithm for deciding if two cyclic words in the standard presentation of the fundamental group of a closed oriented surface of positive genus represent the same conjugacy class. A simple diagram on a surface is a disjoint union of simple closed curves none of which bound a disk. If [Formula: see text] is a once punctured closed surface of negative Euler characteristic, simple diagrams are classified up to isotopy by their geometric intersection numbers with the edges of an ideal triangulation of [Formula: see text]. Simple diagrams on the unpunctured surface [Formula: see text] can be represented by simple diagrams on [Formula: see text]. The weight of a simple diagram is the sum of its geometric intersection numbers with the edges of the triangulation. We show that you can pass from any representative to a least weight representative via a sequence of elementary moves, that monotonically decrease weights. This leads to a geometric analog of Dehn’s algorithm for simple diagrams.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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