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

Koda, Yuya

Algebr. Geom. Topol. 11, No. 1, 417-447 (2011).

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.

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)

Keywords:

knot; unknotting tunnel; complex; toroidal bridge number

References:

[1]	M Boileau, M Rost, H Zieschang, On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces, Math. Ann. 279 (1988) 553 · Zbl 0616.57008
[2]	F Bonahon, J P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. \((4)\) 16 (1983) · Zbl 0545.57002
[3]	S Cho, D McCullough, Cabling sequences of tunnels of torus knots, Algebr. Geom. Topol. 9 (2009) 1 · Zbl 1170.57005
[4]	S Cho, D McCullough, The tree of knot tunnels, Geom. Topol. 13 (2009) 769 · Zbl 1191.57005
[5]	S Cho, D McCullough, Constructing knot tunnels using giant steps, Proc. Amer. Math. Soc. 138 (2010) 375 · Zbl 1192.57004
[6]	S Cho, D McCullough, Tunnel leveling, depth, and bridge numbers, Trans. Amer. Math. Soc. 363 (2011) 259 · Zbl 1210.57004
[7]	H Goda, C Hayashi, Genus two Heegaard splittings of exteriors of \(1\)-genus \(1\)-bridge knots · Zbl 1276.57018
[8]	H Goda, M Scharlemann, A Thompson, Levelling an unknotting tunnel, Geom. Topol. 4 (2000) 243 · Zbl 0958.57007
[9]	C M Gordon, On primitive sets of loops in the boundary of a handlebody, Topology Appl. 27 (1987) 285 · Zbl 0634.57007
[10]	J Hass, A Thompson, W Thurston, Stabilization of Heegaard splittings, Geom. Topol. 13 (2009) 2029 · Zbl 1177.57018
[11]	J Hempel, \(3\)-Manifolds as viewed from the curve complex, Topology 40 (2001) 631 · Zbl 0985.57014
[12]	M Hirasawa, Y Uchida, The Gordian complex of knots, J. Knot Theory Ramifications 11 (2002) 363 · Zbl 1004.57008
[13]	A Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8 (2008) 1403 · Zbl 1151.57007
[14]	A Ishii, K Kishimoto, The IH-complex of spatial trivalent graphs (2009) · Zbl 1213.57010
[15]	J Johnson, Bridge number and the curve complex
[16]	J Johnson, A Thompson, On tunnel number one knots that are not \((1,n)\) · Zbl 1218.57008
[17]	A Kawauchi, A survey of knot theory, Birkhäuser Verlag (1996) · Zbl 0861.57001
[18]	S Kinoshita, On \(\theta_n\)-curves in \(\mathbfR^3\) and their constituent knots (editor S Suzuki), Kinokuniya (1987) 211
[19]	T Kobayashi, Classification of unknotting tunnels for two bridge knots (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2 (1999) 259 · Zbl 0962.57003
[20]	F Luo, On Heegaard diagrams, Math. Res. Lett. 4 (1997) 365 · Zbl 0887.57022
[21]	D McCullough, Virtually geometrically finite mapping class groups of \(3\)-manifolds, J. Differential Geom. 33 (1991) 1 · Zbl 0721.57008
[22]	Y Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988) 465 · Zbl 0651.57012
[23]	K Morimoto, M Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143 · Zbl 0697.57002
[24]	K Morimoto, M Sakuma, Y Yokota, Examples of tunnel number one knots which have the property “\(1+1=3\)”, Math. Proc. Cambridge Philos. Soc. 119 (1996) 113 · Zbl 0866.57004
[25]	M Scharlemann, M Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006) 593 · Zbl 1128.57022
[26]	F Waldhausen, Heegaard-Zerlegungen der \(3\)-Sphäre, Topology 7 (1968) 195 · Zbl 0157.54501