The reductivity of a spherical curve represents how reduced it is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four, that is, we give a set of parts of spherical curves such that every spherical curve with reductivity four has at least one of the parts, by considering 4-gons.
"The Reductivity of Spherical Curves Part II: 4-gons." Tokyo J. Math. 41 (1) 51 - 63, June 2018. https://doi.org/10.3836/tjm/1502179266