The reductivity of spherical curves Part II: 4-gons
  The reductivity of a spherical curve represents how reduced the spherical curve 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 by considering 4-gons. 
Yui Onoda and Ayaka Shimizu
math.GT