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KYUNGPOOK Math. J. 2019; 59(4): 821-834

Published online December 23, 2019

Copyright © Kyungpook Mathematical Journal.

A Note on Unavoidable Sets for a Spherical Curve of Reductivity Four

Kenji Kashiwabara, Ayaka Shimizu∗

Department of General Systems Studies, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8902, Japan
e-mail : kashiwa@idea.c.u-tokyo.ac.jp
Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan
e-mail : shimizu@nat.gunma-ct.ac.jp

Received: October 11, 2018; Revised: March 11, 2019; Accepted: March 18, 2019

Abstract

The reductivity of a spherical curve is the minimal number of times a particular local transformation called an inverse-half-twisted splice is required to obtain a reducible spherical curve from the initial spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configurations for a spherical curve with reductivity four is given by focusing on 5-gons. It has also been unknown if there exists a reduced spherical curve which has no 2-gons and 3-gons of type A, B and C. This paper gives the answer to this question by constructing such a spherical curve.

Keywords: knot projection, reductivity, spherical curve, unavoidable set.