##
**Tunnel complexes of 3-manifolds.**
*(English)*
Zbl 1216.57004

The author defines, for each 3-manifold \(M\) and natural number \(t\), the \(t\)-tunnel complex \({\mathcal T}_t(M)\), whose vertices are knots in \(M\) of tunnel number at most \(t\), and \(k+1\) vertices span a \(k\)-simplex of \({\mathcal T}_t(M)\) if there is a \(\theta\)-curve of \(t+2\) edges in \(M\) such that \(M - \text{Int}((N(\theta))\) is a handlebody and each vertex in the collection corresponds to a constituent knot of \(\theta\). It is shown that for \(M\) a 3-sphere or a lens space, \({\mathcal T}_t(M)\) is connected for all \(t\), and if \(n\) is the number of Heegaard splittings of a closed orientable 3-manifold \(M\) up to isotopy, the number of components of \({\mathcal T}_t(M)\) is at most \(2n\). In case \(M\) is the 3-sphere, it is shown that there is a simplicial surjection from the simplicial complex of Cho and McCullough onto \({\mathcal T}_t(S^3)\).

In a way of generalizing the simplicial complex of Cho and McCullough, the author then considers the complex \(D_t(H_{t+1})\) whose vertices are non-separating \(t\)-tuples of disks in a genus \(t+1\) handlebody \(H_{t+1}\), and \(k+1\) vertices span a \(k\)-simplex if they represent pairwise disjoint disks in \(H_{t+1}\). It is then shown that \(D_t(H_{t+1})\) is connected for all \(t\), and for \(M\) a 3-sphere or lens space and \(H_{t+1}\) a handlebody in \(M\), \(D_t(H_{t+1})\) is connected and there is a surjection from the 1-skeleton of \(D_t(H_{t+1})\) to that of \({\mathcal T}_t(M)\) for all \(t \geq 1\).

The author then defines the \(t\)-distance of two knots of tunnel number at most \(t\) in a closed orientable 3-manifold \(M\) to be the simplicial distance of the corresponding vertices in the 1-skeleton of \({\mathcal T}_t(M)\), and the \(t\)-distance is \(\infty\) if the knots are in different components. It is shown that for every integer \(n\) there is a tunnel number one knot \(K\) in \(S^3\) such that the 1-distance of \(K\) and the unknot is greater than \(n\). It is also shown that if \(K\) is a tunnel number one knot in \(S^3\) whose 1-distance from the unknot is greater than or equal to 2, then \(K\) is hyperbolic.

In a way of generalizing the simplicial complex of Cho and McCullough, the author then considers the complex \(D_t(H_{t+1})\) whose vertices are non-separating \(t\)-tuples of disks in a genus \(t+1\) handlebody \(H_{t+1}\), and \(k+1\) vertices span a \(k\)-simplex if they represent pairwise disjoint disks in \(H_{t+1}\). It is then shown that \(D_t(H_{t+1})\) is connected for all \(t\), and for \(M\) a 3-sphere or lens space and \(H_{t+1}\) a handlebody in \(M\), \(D_t(H_{t+1})\) is connected and there is a surjection from the 1-skeleton of \(D_t(H_{t+1})\) to that of \({\mathcal T}_t(M)\) for all \(t \geq 1\).

The author then defines the \(t\)-distance of two knots of tunnel number at most \(t\) in a closed orientable 3-manifold \(M\) to be the simplicial distance of the corresponding vertices in the 1-skeleton of \({\mathcal T}_t(M)\), and the \(t\)-distance is \(\infty\) if the knots are in different components. It is shown that for every integer \(n\) there is a tunnel number one knot \(K\) in \(S^3\) such that the 1-distance of \(K\) and the unknot is greater than \(n\). It is also shown that if \(K\) is a tunnel number one knot in \(S^3\) whose 1-distance from the unknot is greater than or equal to 2, then \(K\) is hyperbolic.

Reviewer: Michael C. Tsau (St. Louis)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M15 | Relations of low-dimensional topology with graph theory |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

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\textit{Y. Koda}, Algebr. Geom. Topol. 11, No. 1, 417--447 (2011; Zbl 1216.57004)

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