任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
任意に高いヘンペル距離の橋分解 をもつ結び目
市原一裕
日本大学 文理学部
斎藤敏夫氏（上越教育大）との共同研究 2013年度 日本数学会秋期総合分科会
愛媛大学，2013年9月24日
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting
Hemple distance Bridge splitting Result
Heegaard splitting
Heegaard splitting ( Heegaard (1898) )
A decomposition of a closed orientable 3-manifold into two handlebodies.
Moise (1952)
Every closed orientable 3-manifold has a Heegaard splitting.
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance
Bridge splitting Result
Hempel distance
Facts
Certain properties of Heegaard splittings reflect topological characteristics of 3-manifolds:
I any Heegaard splitting of a reducible 3-manifold is reducible (Haken (1968))
I any Heegaard splitting of a non-Haken 3-manifold is reducible or strongly irreducible
(Casson-Gordon (1987)) Motivated by such works, in 2001,
Hempel introduced an invariant of a Heegaard splitting, called the distance, or commonly called the Hempel distance.
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance
Bridge splitting Result
High distance splittings for closed 3-manifolds
A lot of studies have been done about the distance of Heegaard splittings...
Among them, on the existence of high distance splittings, there are several known results.
For example,
Hempel (2001)
There exist Heegaard splittings of closed 3-manifolds with distance at leastn for arbitrarily largen.
(adapting an idea of Kobayashi (1988))
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
Bridge splitting
A natural generalization of Heegaard splitting for a link is given by the bridge splitting(or bridge decomposition).
(g, b)-bridge splitting
A decomposition of(M, L) into
two pairs of a genusg handlebody and btrivial arcs for a link Lin a closed 3-manifoldM.
For bridge splittings, the notion of distance is naturally defined as a generalization of the case of Heegaard splittings.
Precise definition will be given as follows...
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
Curve complex
To define the (Hempel) distance,
we first prepare the terminology about thecurve complex, originally introduced by Harvey (1981).
Let F be a compact orientable surface
possibly with non-empty boundary.
the curve complex C (F )
Thesimplicial complex whose k-simplexes are the isotopy classes of k+ 1collections of mutually non-isotopicessential loops onF which can be realizeddisjointly.
Remark
“essential” means non-trivial and not boundary-parallel.
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
Distance on the curve complex
Distance on the curve complex
For a pair of vertices[x]and[y]in C(F),
thedistanced([x],[y])between[x]and[y]is defined as the minimal number of edges in a path from[x]to[y].
The well-definedness is due to:
Masur-Minsky (1999)
The curve complex is connectedifF is notsporadic,
i.e.,∂F has at least 5 (resp.2) components if g= 0(resp.1).
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
Distance of bridge splitting
For a bridge splitting((V1, t1),(V2, t2))of (M, K), set E(K) :=M −intN(K),
Wi:=Vi∩E(K) , S0:=∂Vi∩E(K) for i= 1,2.
For eachi= 1,2, thedisk setD(Wi) is the set of
vertices corresponding to the curves bounding disks in Wi.
Hempel distance d( D (W
1), D (W
2))
min{d([x],[y]) |[x]∈ D(W1),[y]∈ D(W2)}
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
On high distance bridge splittings
Saito (2004)
In any closed3-manifold with a Heegaard splitting of genus one, there is a knot with a(1,1)-bridge splitting of arbitrary high distance.
Blair-Tomova-Yoshizawa (preprint)
For given integersb, c, g, and n,there existsa c-component linkLin a 3-manifoldM so that(M, L)admits a (g, b)-bridge splitting of distance at leastn.
任意に高いヘンペ ル距離の橋分解を
もつ結び目 K.Ichihara
Heegaard splitting Hemple distance Bridge splitting Result
Main Theorem
Theorem (I.-Saito)
For anygiven closed 3-manifold M with a Heegaard surface S of genus g, and any given positive integersb & n, there exists a knot K in M
admiting a (g, b)-bridge splitting of distance> n with respect to S except for(g, b) = (0,1),(0,2).
