No immersed 2-knot with at most one self-intersection point has triple point number two or three

Abstract

An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in R4, respectively. The triple point number t(F) of an embedded/immersed surface-knot F is the minimum number of triple points required for a diagram of F. Satoh proved that (i) there does not exist an embedded surface-knot F such that t(F)=1, and (ii) there does not exist an embedded 2-knot F such that t(F)=2 or 3. In this paper, we prove similar results for immersed surface-knots with some conditions.