Connected sums of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml125" display="inline" overflow="scroll" altimg="si125.gif"><mml:mi>z</mml:mi></mml:math>-knotted triangulations

Pankov, Mark; Tyc, Adam

doi:10.1016/j.ejc.2018.02.010

Cited by 7 publications

(9 citation statements)

References 8 publications

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“…If k is even, then this zigzag is See [15,16] for more examples of z-knotted triangulations. Examples of z-knotted fullerenes can be found in [7].…”

Section: Zigzags and Z-orientations Of Triangulations Of Surfacesmentioning

confidence: 99%

“…If we replace a z-orientation by the reversed z-orientation, then the type of every edge does not change (but all edges of type II reverse the directions), consequently, the types of vertices and faces also do not change. For z-knotted triangulations there is a unique z-orientation (up to reversing) and we can determine the types of edges, vertices and faces without attaching to a z-orientation [15].…”

Section: Zigzags and Z-orientations Of Triangulations Of Surfacesmentioning

confidence: 99%

“…spherical triangulations, have the same zigzag structure. Zigzags in triangulations of surfaces (not necessarily orientable) are investigated in [15,16,17]. By [16], every such triangulation admits a z-knotted shredding, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Z-oriented triangulations of surfaces

Tyc

2022

Ars Math. Contemp.

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The main objects of the paper are z-oriented triangulations of connected closed 2dimensional surfaces. A z-orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge -zigzags from the z-orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate z-oriented triangulations with all faces of the first type (in the general case, any z-oriented triangulation can be shredded to a z-oriented triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of type I after any edge of type II. We give a topological characterization of the homogeneity of zigzags; in particular, we describe a one-to-one correspondence between z-oriented triangulations with homogeneous zigzags and closed 2-cell embeddings of directed Eulerian graphs in surfaces. At the end, we give an application to one type of the z-monodromy.

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“…If k is even, then this zigzag is See [15,16] for more examples of z-knotted triangulations. Examples of z-knotted fullerenes can be found in [7].…”

Section: Zigzags and Z-orientations Of Triangulations Of Surfacesmentioning

confidence: 99%

Section: Zigzags and Z-orientations Of Triangulations Of Surfacesmentioning

confidence: 99%

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Z-oriented triangulations of surfaces

Tyc

2022

Ars Math. Contemp.

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“…In fact, zigzags of such embedded graphs are geometrical realizations of Gauss codes. Infinite series of z-knotted triangulations for any closed (not necessarily) oriented surfaces were constructed in [8].…”

Section: Introductionmentioning

confidence: 99%

$Z$-knotted and $Z$-homogeneous triangulations of surfaces

Tyc¹

2020

Preprint

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A triangulation is called z-knotted if it has a single zigzag (up to reversing). A z-orientation on a triangulation is a minimal collection of zigzags which double covers the set of edges. An edge is of type I if zigzags from the z-orientation pass through it in different directions, otherwise this edge is of type II. If all zigzags from the z-orientation contain precisely two edges of type I after any edge of type II, then the z-oriented triangulation is said to be z-homogeneous. We describe an algorithm transferring each z-homogeneous trianguation to other z-homogeneous triangulation which is also z-knotted.

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Section: Introductionmentioning

confidence: 99%

Triangulations with homogeneous zigzags

Kwiatkowski,

Pankov,

Tyc

2019

Preprint

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We investigate zigzags in triangulations of connected closed 2dimensional surfaces and show that there is a one-to-one correspondence between triangulations with homogeneous zigzags and closed 2-cell embeddings of directed Eulerian graphs in surfaces. A triangulation is called z-knotted if it has a single zigzag. We construct a family of tree structured z-knotted spherical triangulations whose zigzags are homogeneous.

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Connected sums of z-knotted triangulations

Cited by 7 publications

References 8 publications

Z-oriented triangulations of surfaces

Z-oriented triangulations of surfaces

$Z$-knotted and $Z$-homogeneous triangulations of surfaces

Triangulations with homogeneous zigzags

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