Kirby-Thompson distance for trisections of knotted surfaces. (English) Zbl 1541.57021
A bridge trisection \(\mathcal{T}\) of a closed surface \(S\) in \(S^4\) is a decomposition \((S^4,S)=(X_1, \mathcal{D}_1) \cup (X_2, \mathcal{D}_2) \cup (X_3, \mathcal{D}_3)\), where each \(X_i\) is a 4-ball, and each \(\mathcal{D}_i=X_i \cap S\) is a collection of several boundary-parallel disks in \(X_i\), and satisfying some other conditions. The bridge number \(b(\mathcal{T})\) of \(\mathcal{T}\) is the number \(|S\cap (X_1 \cap X_2 \cap X_3)|/2\), and the bridge number \(b(S)\) of \(S\) is the minimum of the bridge numbers \(b(\mathcal{T})\) over all bridge trisections \(\mathcal{T}\) of \(S\). A bridge trisection is defined for a smooth surface in a closed 4-manifold, and every surface embedded in a closed 4-manifold has a bridge trisection.
In this paper, the authors give an integer invariant \(\mathcal{L}(\mathcal{T})\) of a bridge trisection \(\mathcal{T}\) of a properly embedded surface \(S\) in \(S^4\) or \(B^4\), and they define an invariant \(\mathcal{L}(S)\) as the minimum of \(\mathcal{L}(\mathcal{T})\) over all trisections \(\mathcal{T}\) of \(S\) with \(b(\mathcal{T})=b(S)\); the invariant \(\mathcal{L}(S)\) resembles the Kirby-Thompson invariant \(\mathcal{L}(X)\) of a smooth 4-manifold \(X\).
The main results are as follows. A smooth closed surface \(S\) in \(S^4\) with \(\mathcal{L}(S)=0\) is the distant sum of unknotted 2-spheres and unknotted nonorientable surfaces. The authors give a lower bound of \(\mathcal{L}(S)\) of a smooth closed surface \(S\) in \(S^4\) when \(S\) is connected, orientable and irreducible, in terms of the bridge number \(b(S)\) and the genus \(g(S)\) of \(S\), and they give another lower bound of \(\mathcal{L}(S)\) when \(S\) is prime, in terms of \(b(S)\) and the Euler characteristic \(\chi(S)\). Here, a connected and orientable surface \(S\) is irreducible if it is nontrivial and not the connected sum of a nontrivial surface and a trivial surface of positive genus, and \(S\) is prime if it is nontrivial and not a connected sum or distant sum of nontrivial smooth surfaces.
The authors describe how bridge surfaces and trisection surfaces can be cut open along certain spheres, and how this operation is related with connected sums and distant sums. And they define the invariant \(\mathcal{L}\) using the pants complex and its distance.
In this paper, the authors give an integer invariant \(\mathcal{L}(\mathcal{T})\) of a bridge trisection \(\mathcal{T}\) of a properly embedded surface \(S\) in \(S^4\) or \(B^4\), and they define an invariant \(\mathcal{L}(S)\) as the minimum of \(\mathcal{L}(\mathcal{T})\) over all trisections \(\mathcal{T}\) of \(S\) with \(b(\mathcal{T})=b(S)\); the invariant \(\mathcal{L}(S)\) resembles the Kirby-Thompson invariant \(\mathcal{L}(X)\) of a smooth 4-manifold \(X\).
The main results are as follows. A smooth closed surface \(S\) in \(S^4\) with \(\mathcal{L}(S)=0\) is the distant sum of unknotted 2-spheres and unknotted nonorientable surfaces. The authors give a lower bound of \(\mathcal{L}(S)\) of a smooth closed surface \(S\) in \(S^4\) when \(S\) is connected, orientable and irreducible, in terms of the bridge number \(b(S)\) and the genus \(g(S)\) of \(S\), and they give another lower bound of \(\mathcal{L}(S)\) when \(S\) is prime, in terms of \(b(S)\) and the Euler characteristic \(\chi(S)\). Here, a connected and orientable surface \(S\) is irreducible if it is nontrivial and not the connected sum of a nontrivial surface and a trivial surface of positive genus, and \(S\) is prime if it is nontrivial and not a connected sum or distant sum of nontrivial smooth surfaces.
The authors describe how bridge surfaces and trisection surfaces can be cut open along certain spheres, and how this operation is related with connected sums and distant sums. And they define the invariant \(\mathcal{L}\) using the pants complex and its distance.
Reviewer: Inasa Nakamura (Saga)
MSC:
| 57K45 | Higher-dimensional knots and links |
References:
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