Plane curves in an immersed graph in R2

Marisa Sakamoto; Kouki Taniyama

doi:10.1142/S021821651350003X

Plane curves in an immersed graph in R²

Marisa Sakamoto^*, Kouki Taniyama

^*Corresponding author for this work

School of Education

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

Original language	English
Article number	1350003
Journal	Journal of Knot Theory and its Ramifications
Volume	22
Issue number	2
DOIs	https://doi.org/10.1142/S021821651350003X
Publication status	Published - 2013 Feb

Keywords

Immersed graph
chord diagram
knot
plane curve

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1142/S021821651350003X

Cite this

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title = "Plane curves in an immersed graph in R2",

abstract = "For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.",

keywords = "Immersed graph, chord diagram, knot, plane curve",

author = "Marisa Sakamoto and Kouki Taniyama",

note = "Funding Information: The authors are grateful to Professors Shin{\textquoteright}ichi Suzuki and Toshie Takata for their encouragements. The authors are also grateful to Professors Toshiki Endo, Reiko Shinjo, Noboru It{\^o} and Ryo Hanaki for their helpful comments. The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 24540100), Japan Society for the Promotion of Science.",

year = "2013",

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doi = "10.1142/S021821651350003X",

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AU - Sakamoto, Marisa

AU - Taniyama, Kouki

N1 - Funding Information: The authors are grateful to Professors Shin’ichi Suzuki and Toshie Takata for their encouragements. The authors are also grateful to Professors Toshiki Endo, Reiko Shinjo, Noboru Itô and Ryo Hanaki for their helpful comments. The second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 24540100), Japan Society for the Promotion of Science.

PY - 2013/2

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N2 - For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

AB - For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

KW - Immersed graph

KW - chord diagram

KW - knot

KW - plane curve

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