Bridge trisections of knotted surfaces in 4-manifolds. (English) Zbl 1418.57017
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.
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.
Reviewer: Inasa Nakamura (Kanazawa)
MSC:
57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |