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 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).

scholarly journals Centralisers of Dehn twist automorphisms of free groups

2015 ◽  
Vol 159 (1) ◽  
pp. 89-114 ◽  
Author(s):  
MORITZ RODENHAUSEN ◽  
RICHARD D. WADE
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Dehn Twist ◽  
Classifying Space ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups

AbstractWe refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

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 Generating the twist subgroup by involutions

2020 ◽  
pp. 1-24
Author(s):  
Tüli̇n Altunöz ◽  
Mehmetci̇k Pamuk ◽  
Oguz Yildiz
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Index Formula ◽  
Mapping Class ◽  
New Techniques ◽  
Closed Curves ◽  
Nonorientable Surface ◽  
Dehn Twists ◽  
Generating Sets ◽  
Number Of Generators

For a nonorientable surface, the twist subgroup is an index [Formula: see text] subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

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.

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 Parity criterion for unstabilized Heegaard splittings

2010 ◽  
Vol 149 (1) ◽  
pp. 115-125
Author(s):  
JUNG HOON LEE
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Branched Coverings ◽  
Dehn Twists ◽  
Heegaard Diagram ◽  
Parity Condition

AbstractWe give a parity condition of a Heegaard diagram implying that it is unstabilized. As applications, we show that Heegaard splittings of 2-fold branched coverings of n-component, n-bridge links in S3 are unstabilized, and we also construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.

scholarly journals Quotient and blow-up of automorphisms of graphs of groups

2018 ◽  
Vol 28 (05) ◽  
pp. 733-758
Author(s):  
Kaidi Ye
Keyword(s):  
Blow Up ◽  
Correction Term ◽  
Free Groups ◽  
Dehn Twist ◽  
Graph Of Groups ◽  
Graphs Of Groups ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups ◽  
Automorphisms Of Graphs

In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

Properties of Casson–Gordon’s rectangle condition

2020 ◽  
Vol 29 (12) ◽  
pp. 2050083
Author(s):  
Bo-Hyun Kwon ◽  
Jung Hoon Lee
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Diagram ◽  
Strongly Irreducible ◽  
Strong Irreducibility

For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

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