Let \(X\) be a smooth, closed, connected, orientable 4-manifold. A trisection \(\mathcal{T}\) of \(X\) is a decomposition \(X=X_1 \cup X_2 \cup X_3\), where \(X_i\) is a 1-handlebody, \(H_{ij}=X_i \cap X_j\) is a handlebody for \(i \neq j\), and \(\Sigma= X_1 \cap X_2 \cap X_3\) is a closed surface. Here, a handlebody (respectively, a 1-handlebody) is the boundary-connected sum of several copies of \(S^1 \times D^2\) (respectively, \(S^1 \times B^3\)). A knotted surface in \(X\) is a smoothly embedded, closed surface. It is known that a knotted surface in \(S^4\) admits a special trisection called a bridge trisection, which is a generalization of a bridge splitting of a knot in \(S^3\).
In the paper under review, the authors give generalized notions of a bridge trisection and bridge position as follows. Let \(X\) be a smooth, closed, connected, orientable 4-manifold, and let \(K\) be a knotted surface in \(X\). A generalized bridge trisection of \((X, K)\) is a decomposition \((X, K)=(X_1, D_1) \cup (X_2, D_2) \cup (X_3, D_3)\), where \(X=X_1 \cup X_2 \cup X_3\) is a trisection, \(D_i\) consists of “trivial disks” in \(X_i\), and \(\tau_{ij}= D_i \cap D_j\) \((i \neq j)\) consists of arcs forming a “trivial tangle” in \(H_{ij}\). For a trisection \(\mathcal{T}\) of \(X\) given by \(X=X_1 \cup X_2 \cup X_3\) and a knotted surface \(K\) in \(X\), \(K\) is in bridge position with respect to \(\mathcal{T}\) if \((X, K)=(X_1, K \cap X_1) \cup (X_2, K \cap X_2) \cup (X_3, K \cap X_3)\) is a generalized bridge trisection. For a trivial tangle \(\tau\) in a handlebody \(H\), a curve-and-arc system determining \((H, \tau)\) is a collection of pairwise disjoint simple closed curves and arcs in \(\partial H\) determined from \(H\) and \(\tau\). A shadow diagram for a generalized bridge trisection \(\mathcal{T}\) is a triple of curve-and-arc systems obtained from \(\mathcal{T}\). A \((g,1)\)-generalized bridge trisection is a generalized bridge trisection such that the central surface \(\Sigma= X_1 \cap X_2 \cap X_3\) is of genus \(g\) and meets a knotted surface \(K\) in 2 points. In this case, the associated trisection is a 1-bridge trisection and its bridge position is called 1-bridge position.
The main results are as follow. For a smooth, closed, connected, orientable 4-manifold \(X\) with a trisection \(\mathcal{T}\), any knotted surface in \(X\) can be isotoped into bridge position with respect to \(\mathcal{T}\). A generalized bridge trisection of a knotted surface \(K\) in \(X\) induces a shadow diagram, and a generalized bridge trisection is determined by its shadow diagram. In particular, if \(K\) is a 2-knot, then \(K\) can be isotoped into 1-bridge position and \((X, K)\) admits a doubly-pointed trisection diagram, which is a shadow diagram where each curve-and-arc-system contains exactly one arc. Here, a 2-knot is an embedded 2-sphere. The authors consider the 1-bridge trisections (\((g,1)\)-generalized bridge trisections) and give a classification of \((1,1)\)-generalized bridge trisections. They give results concerning generalized bridge trisections of complex curves in complex 4-manifolds of low trisection genus. In particular, they show the following: Let \(X_{n,d}\) be the 4-manifold obtained as the \(n\)-fold cover of \(\mathbb{C}P^2\) branched along the complex curve of degree \(d\). Then, \(X_{n,d}\) admits an efficient \((g,0)\)-trisection for \(g\) determined from \(n\) and \(d\). Note that \(X_{n,d}\) form a set including exotic 4-manifolds. The paper is closed by discussing and giving a conjecture on uniqueness of generalized bridge trisections. In the proof, the authors discuss deformations using Morse functions and meridional stabilizations.