On a banded link presentation of knotted surfaces (Q2799078)

A knotted surface is the image of an embedding of a closed surface into \(\mathbb R^4\), and in particular, we call it a surface-knot when it is connected. In this paper, the author investigates knotted surfaces in \(\mathbb R^4\), by using a banded link presentation. Any knotted surface \(F\) is deformed by equivalence to the following form: (i) \(F\) has only finitely many Morse critical points, (ii) all maximal points lie in \(\mathbb R^3 \times\{1\}\), (iii) all minimal points lie in \(\mathbb R^3 \times\{-1\}\), and (iv) all saddle points lie in \(\mathbb R^3\times \{0\}\); see [\textit{S. J. Lomonaco jun.}, Pac. J. Math. 95, 349--390 (1981; Zbl 0483.57012)] and [\textit{A. Kawauchi} et al., Math. Semin. Notes, Kobe Univ. 10, 75--126 (1982; Zbl 0506.57014)]. The intersection \(F \cap (\mathbb R^3 \times \{0\})\) is deformed to a union of a link \(L \subset \mathbb R^3\) and a finite set \(B\) of pairwise disjoint bands spanning \(L\), and the pair \((L, B)\) is called a banded link. A banded link associated with a knotted surface gives us enough information to obtain the original knotted surface.NEWLINENEWLINEIn this paper, the author mainly considers surface-knots, and introduces the notion of the band number of a surface-knot \(F\) and the long flat form of a banded link associated with \(F\), and shows the following results. The band number of a surface-knot \(F\), denoted by \(\mathrm{bn}(F)\), is the minimum number of bands in a banded link among all banded links which present \(F\). If a surface-knot \(F\) satisfies \(\mathrm{bn}(F)=1\), then \(F\) is the unknotted positive or negative projective plane; this implies that if a banded link presenting a surface-knot has one band, then the surface-knot is unknotted. For the \(n\)-twist-spun \(F\) of a nontrivial 2-bridge classical knot with \(n \neq \pm 1\), \(\mathrm{bn}(F)=2\). For any non-negative integer \(n\), there exists a surface-knot \(F\) with \(\mathrm{bn}(F)=n\). A banded link \((L, B) \subset \mathbb{R}^3\) associated with a knotted surface is in flat form if the link \(L\) is a split link embedded in \(\mathbb{R}^2 \subset \mathbb{R}^3\), and in particular, a banded link associated with a surface-knot is in long flat form if moreover the bands satisfy certain conditions. For example, the author gives diagrammatically the long flat form of the \(n\)-twist-spun \((2, 2k+1)\)-torus knot for integers \(n\) and \(k\).