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Funakoshi, Yukari; Noguchi, Kenta; Shimizu, Ayaka

Unknottability of spatial graphs by region crossing changes. (English) Zbl 1533.57004

Osaka J. Math. 60, No. 3, 671-682 (2023).
In 2010, K. Kishimoto in a seminar at Osaka City University defined region crossing change for knots, which is a local transformation that changes all the crossings on the boundary of a region. In [J. Math. Soc. Japan 66, No. 3, 693–708 (2014; Zbl 1297.57021)], A. Shimizu showed that region crossing change is an unknotting operation for knots. In [J. Knot Theory Ramifications 24, No. 8, Article ID 1550045, 12 p. (2015; Zbl 1336.57008)], K. Hayano et al. conducted a study on region crossing changes on spatial graph diagrams.
In the paper under review, the authors give a necessary and sufficient condition that a spatial graph of a planar graph is unknottable by region crossing changes, and that a spatial graph is completely splitted by region crossing changes. Unlike knots, graphs have vertices. The authors focus on odd degree vertices. The main idea is to use vertices of odd degree to perform an unknotting operation through region crossing changes. For anyone familiar with region crossing changes on knots, Fig. 6 and Fig. 7 in the paper will give the idea.
Reviewer: Yoshiyaki Uchida (Kobe)

MSC:

57K10 Knot theory
57M15 Relations of low-dimensional topology with graph theory

Keywords:

planar graph; spatial graph; Eulerian graph; completely splittable; region crossing change

Citations:

Zbl 1297.57021; Zbl 1336.57008
PDFBibTeX XMLCite
\textit{Y. Funakoshi} et al., Osaka J. Math. 60, No. 3, 671--682 (2023; Zbl 1533.57004)
Full Text: arXiv Link

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