Bounds for Kirby-Thompson invariants of knotted surfaces (Q6053479)

A surface-link is a closed surface smoothly embedded in \(S^4\). For a surface-link \(F\) in \(S^4\), a \((b; c_1, c_2, c_3)\)-bridge trisection \(\mathcal{T}\) is a decomposition \((S^4, F)=(B^4, \mathcal{D}_1) \cup (B^4, \mathcal{D}_2) \cup (B^4, \mathcal{D}_3)\), where \(\mathcal{D}_i\) is a collection of \(c_i\) disks properly embedded in \(B^4\) \((i=1,2,3)\), satisfying certain conditions and \((B^4, \mathcal{D}_i) \cap (B^4, \mathcal{D}_j)\) \((i \neq j)\) is a \(b\)-strand trivial tangle. In particular, a \((b; c, c, c)\)-bridge trisection is called a \((b;c)\)-bridge trisection. \newline The main results are as follows: The authors introduce the notion of the \(\mathcal{L}^*\)-invariant \(\mathcal{L}^*(\mathcal{T})\) for a bridge trisection \(\mathcal{T}\). The \(\mathcal{L}^*\)-invariant \(\mathcal{L}^*(\mathcal{T})\) is an analog to the \(\mathcal{L}\)-invariant \(\mathcal{L}(\mathcal{T})\) due to \textit{R. Blair} et al. [J. Lond. Math. Soc., II. Ser. 105, No. 2, 765--793 (2022; Zbl 07731377)] one dimension lower, defined by using the dual curve complex, while \(\mathcal{L}(\mathcal{T})\) is defined by using the pants complex; in particular, \(\mathcal{L}(\mathcal{T}) \geq \mathcal{L}^*(\mathcal{T})\). The authors give lower bounds for the \(\mathcal{L}^*\)-invariant and the \(\mathcal{L}\)-invariant of a \((b;c)\)-bridge trisection, in terms of \(b\) and \(c\). The authors determine the form of a bridge trisection \(\mathcal{T}\) with \(\mathcal{L}^*(\mathcal{T}) \leq 2\); in particular, it follows that the underlying surface-link is unknotted. The authors give several examples for which the \(\mathcal{L}^*\)-invariant or the \(\mathcal{L}\)-invariant is determined. Further, they discuss the \(\mathcal{L}^*\)-invariant for knotted surfaces with boundary. In order to show the results, they investigate the dual curve complex and the pants complex.