Spatial graphs, tangles and plane trees

Nezhinskiĭ, V.

doi:10.1090/spmj/1633

Spatial graphs, tangles and plane trees
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by V. M. Nezhinskiĭ
Translated by: V. M. Nezhinskiĭ

St. Petersburg Math. J. 31 (2020), 1055-1063

DOI: https://doi.org/10.1090/spmj/1633

Published electronically: October 27, 2020

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Abstract:

All (finite connected) spatial graphs are supplied with an additional structure — the replenished skeleton and its disk framing, — in such a way that the problem of isotopic classification of spatial graphs endowed with this structure admits reduction to two problems: the (classical) problem of isotopic classification of tangles and the (close to classical) problem of isotopic classification of plane trees equipped with an additional structure, specifically, a set of hanging vertices and a fixed vertex (the root of the tree) in this set.

References

Yu. V. Maslova and V. M. Nezhinskiĭ, Framings of maximal trees by pairs of chords, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 415 (2013), no. Geometriya i Topologiya. 12, 91–102 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 212 (2016), no. 5, 577–583. MR 3473018
Hiromasa Moriuchi, A table of $\theta$-curves and handcuff graphs with up to seven crossings, Noncommutativity and singularities, Adv. Stud. Pure Math., vol. 55, Math. Soc. Japan, Tokyo, 2009, pp. 281–290. MR 2463504, DOI 10.2969/aspm/05510281
Hiromasa Moriuchi, Enumeration of algebraic tangles with applications to theta-curves and handcuff graphs, Kyungpook Math. J. 48 (2008), no. 3, 337–357. MR 2454677, DOI 10.5666/KMJ.2008.48.3.337
Hiromasa Moriuchi, An enumeration of theta-curves with up to seven crossings, J. Knot Theory Ramifications 18 (2009), no. 2, 167–197. MR 2507922, DOI 10.1142/S0218216509006884
V. M. Nezhinskiĭ and Yu. V. Maslova, Links of graphs with framed vertices, Vestn. S.-Peterburg Univ. Ser. Mat. Mekh. Astronom. (2012), no. 2, 57–60. (Russian)
V. M. Nezhinskiĭ and Yu. V. Maslova, Vertex-framed graphs, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 415 (2013), no. Geometriya i Topologiya. 12, 103–108 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 212 (2016), no. 5, 584–586. MR 3473019
V. M. Nezhinskiĭ and Yu. V. Maslova, Framings of spatial graphs, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 464 (2017), no. Kombinatorika i Teoriya Grafov. IX, 88–94 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 236 (2019), no. 5, 527–531. MR 3748112, DOI 10.1007/s10958-018-4130-4
Natsumi Oyamaguchi, Enumeration of spatial 2-bouquet graphs up to flat vertex isotopy. part B, Topology Appl. 196 (2015), no. part B, 805–814. MR 3431017, DOI 10.1016/j.topol.2015.05.049
Richard S. Palais, Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 274–277. MR 117741, DOI 10.1090/S0002-9939-1960-0117741-0
Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362