Minimal triple point numbers of some non-orientable surface-links. (Q5930565)

A surface-link \(F\) is a closed surface (possibly disconnected) smoothly embedded in the Euclidean 4-space \(\mathbb R^4\). Two surface-links \(F\) and \(F'\) are equivalent if there is an ambient isotopy of \(\mathbb R^4\) that maps \(F\) onto \(F'\). A projection \(\mathbb R^4\to \mathbb R^3\) is called generic with respect to a surface-link \(F\) if its restriction onto \(F\) is a generic map \(F\to \mathbb R^3\). The triple point number \(t(F)\) of surface-links which is defined as the minimal number of triple points in a generic projection of \(F\) into \(\mathbb R^3\). The main result of the paper states: \textbf{Theorem:} \textit{For any positive integer \(N\) there exists a 2-component surface-link \(F=F_1\cup F_2\) such that (i) each \(F_i\) is a non-orientable surface-knot, (ii) \(\chi(F_i) = 2-N\), (\(i=1, 2\)), (iii) \(e(F_1) = 2N\) and \(e(F_2) = -2N\), (iv) \(\pi_1(\mathbb R^4- F)\simeq \langle a, b| aba=b, bab=a\rangle\), (v) \(t(F) = 2N\), where \(\chi\) denotes the Euler characteristic, and \(e\) denotes the normal Euler number.}