Roots of crosscap slides and crosscap transpositions

Parlak, Anna; Stukow, Michał

doi:10.1007/s10998-017-0210-3

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…Remark 4.6. Using a computer it can be checked that there are 59 values of g ≤ 500 satisfying the assumptions of case (11) and for 31 of these values N = g .…”

Section: Degrees Of Roots Of Dehn Twistsmentioning

confidence: 99%

“…. , n − 1 (for a definition see for example Section 2 of [11]) and let η i = t −1 c i+1 t z i u 1,i u 2,i · · · u n−1,i , for i = 1, . .…”

Section: Expressions For Primary Rootsmentioning

confidence: 99%

“…They also gave an arithmetic description of conjugacy classes of such roots. Our aim is to conduct an analogous investigation in the nonorientable case, which is a natural continuation of our previous work in [11].…”

Section: Introductionmentioning

confidence: 97%

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Roots of Dehn twists on nonorientable surfaces

Parlak

Stukow

2019

J. Knot Theory Ramifications

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Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist tc about a nonseparating circle c in the mapping class group M(Ng) of a nonorientable surface Ng of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n > 1, then for each sufficiently large g, tc has a root of degree n in M(Ng). Moreover, for any possible degree n we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in Ng.

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“…Remark 4.6. Using a computer it can be checked that there are 59 values of g ≤ 500 satisfying the assumptions of case (11) and for 31 of these values N = g .…”

Section: Degrees Of Roots Of Dehn Twistsmentioning

confidence: 99%

“…. , n − 1 (for a definition see for example Section 2 of [11]) and let η i = t −1 c i+1 t z i u 1,i u 2,i · · · u n−1,i , for i = 1, . .…”

Section: Expressions For Primary Rootsmentioning

confidence: 99%

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Roots of Dehn twists on nonorientable surfaces

Parlak

Stukow

2019

J. Knot Theory Ramifications

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“…Crosscap transpositions appear already in the proof of the main result of [2], and their name was introduced in [19]. Recently Parlak and Stukow [14] proved that in M(N g,0 ) crosscap slides have nontrivial roots if g ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

Generating the mapping class group of a nonorientable surface by crosscap transpositions

Leśniak

Szepietowski

2017

Topology and its Applications

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A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus g ≥ 7 is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of g + 2 crosscap transpositions generating the mapping class group.

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“…For an illustration of this and the following transformations, refer to [37]. The Dehn twist T ζ has the representation…”

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confidence: 99%

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

Kolbe¹,

Evans²

2018

Preprint

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We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group in the group of outer automorphisms of the symmetry group of a tiling. We explicitly give descriptions of certain subgroups of mapping class groups and of tilings as decorations on orbifolds, namely those that are commensurate with the Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this explicit description to give an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations, outlining how the approach yields an unambiguous enumeration

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Roots of crosscap slides and crosscap transpositions

Cited by 4 publications

References 14 publications

Roots of Dehn twists on nonorientable surfaces

Roots of Dehn twists on nonorientable surfaces

Generating the mapping class group of a nonorientable surface by crosscap transpositions

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

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