Embeddings of Four-valent Framed Graphs into 2-surfaces

Manturov, Vassily Olegovich

doi:10.1007/978-3-642-15637-3_7

Cited by 9 publications

(13 citation statements)

References 26 publications

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“…Our methods are close to those used in our previous paper [3], which were themselves based closely on those used by the second named author (V.O.M.) in [5] for framed four-valent graphs.…”

supporting

confidence: 59%

Checkerboard embeddings of *-graphs into nonorientable surfaces

Friesen

Manturov

2014

J. Knot Theory Ramifications

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This paper considers * -graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of * -graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given * -graph in which all vertices have degree 4 or 6 admits a Z 2 -homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a * -graph with all vertices of degree 4 or 6 admits a Z 2 -homologically trivial embedding into the projective plane or into the Klein bottle.

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“…Our methods are close to those used in our previous paper [3], which were themselves based closely on those used by the second named author (V.O.M.) in [5] for framed four-valent graphs.…”

supporting

confidence: 59%

Checkerboard embeddings of *-graphs into nonorientable surfaces

Friesen

Manturov

2014

J. Knot Theory Ramifications

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“…This structure was called an alternating orientation for a graph (in the present work, we call this graph a frame of an atom). Proposition 1.1 (see [171]). The frame of an atom admits a source-sink structure if and only if the atom is orientable.…”

Section: Atoms and Virtual Knotsmentioning

confidence: 95%

“…As it turned out further, atoms played an important role in knot theory and low-dimensional topology, see [136,147,151,171] and references therein. The graph Γ is said to be the frame of the atom.…”

Section: Atoms and Virtual Knotsmentioning

confidence: 99%

Parity in knot theory and graph-links

2013

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The present monograph is devoted to low-dimensional topology in the context of two thriving theories: parity theory and theory of graph-links, the latter being an important generalization of virtual knot theory constructed by means of intersection graphs. Parity theory discovered by the second-named author leads to a new perspective in virtual knot theory, the theory of cobordisms in two-dimensional surfaces, and other new domains of topology. Theory of graph-links highlights a new combinatorial approach to knot theory.

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“…In topology and graph theory, many notions often have their "odd", "non-orientable", "framed" counterparts, see for example [6,10,11,13,17,18,19,20,21,22,23,24]. Usually, even objects are better understood, however, in the odd case, it is much easier to catch the non-trivial information.…”

Section: Introductionmentioning

confidence: 99%

A parity map of framed chord diagrams

Ilyutko

Manturov

2015

J. Knot Theory Ramifications

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We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In the present paper, we first define a new module M 2 which is generated by chord diagrams on two circles and factored by 4T-relations. Then we construct a "parity" map from the module of framed chord diagrams into M 2 and a weight system on M 2 . Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

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Embeddings of Four-valent Framed Graphs into 2-surfaces

Cited by 9 publications

References 26 publications

Checkerboard embeddings of *-graphs into nonorientable surfaces

Checkerboard embeddings of *-graphs into nonorientable surfaces

Parity in knot theory and graph-links

A parity map of framed chord diagrams

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