Triple point invariants of non-orientable surface-links (Q1612204)

A surface-link \(F=F_1\cup F_2\cup \dots \cup F_n\) is a locally flat 2-submanifold of \(\mathbb{R}^4\), where each \(F_i\) is homeomorphic to a closed surface. A surface-link \(F\) is generic with respect to a projection \(\mathbb{R}^4 \to \mathbb{R}^3\) if its double point set consists of isolated branch points, double point curves and isolated triple points. A triple point is said to be of type \((p,q,r)\) if, with respect to the projection direction, \(F_p\) is the top sheet, \(F_q\) is the middle sheet, and \(F_r\) is the bottom sheet. \(N(p,q, r)\) denotes the number of triple points of type \((p,q,r)\) on a generic projection of \(F\). The author proves in the paper that the numbers \(N(p,q,r)\pmod 2\) are isotopy invariants of \(F\) if \(p\not=q\not=r\). Another result states that \(N(p,q,p)=N(q,p,q) \pmod 2\) and, moreover, \(N(p,q,p)=N(q,p,q) =0 \pmod 2\) if either \(F_p\) or \(F_q\) is orientable. The paper gives also a homological interpretation of the numbers \(N(p,q,p)\pmod 2\) in the case when the surfaces \(F_p\) and \(F_q\) are nonorientable.