Grained manifold

Grained manifolds (engl .: "sutured manifolds") is a term from the mathematical field of 3-dimensional topology , in particular in the construction firmer Foliations and in the calculation of node sex is used.

definition

A grained manifold is a compact oriented 3-manifold with a decomposition of the boundary

${\displaystyle \partial M=\partial _{\tau }M\cup \partial _{\pitchfork }M}$,

in which

• ${\displaystyle \partial _{\pitchfork }M=A\cup T}$consists of unions of circular cylinders and tori and ${\displaystyle A}$ ${\displaystyle T}$
• contains the interior of each circular cylinder in a homologically nontrivial curve (the "scar"), and ${\displaystyle A}$
• all components of are oriented. ${\displaystyle \partial _{\tau }M}$

One denotes with or the associations of the components of whose orientations agree or do not agree with those of . ${\displaystyle R_{+}}$ ${\displaystyle R_{-}}$ ${\displaystyle \partial _{\tau }M}$ ${\displaystyle \partial M}$

A grained manifold is called tight if it is irreducible and every component of the Thurston norm minimizes in its homology class . ${\displaystyle M}$ ${\displaystyle \partial _{\tau }M}$

A grained manifold is called a grained product if it is of the same shape as for a surface . ${\displaystyle M=S\times \left[0,1\right]}$ ${\displaystyle A=\partial S\times \left[0,1\right]}$ ${\displaystyle S}$

Decomposition of grained manifolds

Let be a grained manifold and an actually embedded surface, the intersection of which with each component of either an actually embedded interval or a simple closed curve homologous to a scar or a homologically nontrivial closed curve in a torus component, whereby no more than one homologous to each other Curves may occur as cuts with a torus component. Then ${\displaystyle M}$ ${\displaystyle S\subset M}$ ${\displaystyle \partial _{\pitchfork }M}$

${\displaystyle M^{\prime }:=M\setminus N(S)}$

(the complement of a tube environment of ) also has a grained manifold ${\displaystyle S}$

${\displaystyle \partial _{\pitchfork }M^{\prime }:=(\partial _{\pitchfork }M\cap M^{\prime })\cup N(S_{+}\cap R_{-})\cup N(S_{-}\cap R_{+})}$

${\displaystyle R_{+}^{\prime }:=((R_{+}\cap M^{\prime })\cup S_{+})\setminus int(\partial _{\pitchfork }M^{\prime })}$

${\displaystyle R_{-}^{\prime }:=((R_{+}\cap M^{\prime })\cup S_{-})\setminus int(\partial _{\pitchfork }M^{\prime })}$,

where the two are copies of in . ${\displaystyle S_{+},S_{-}}$ ${\displaystyle S}$ ${\displaystyle \partial M^{\prime }=\partial (M\setminus N(S))}$

A grained manifold is said to be decomposable if there is a sequence

${\displaystyle M_{1},M_{2},\ldots ,M_{n}}$

gives so that a product and each is a decomposition of . The result is grained variety Hierarchy (ger .: sutured manifold hierarchy ). ${\displaystyle M_{1}=M}$ ${\displaystyle M_{n}}$ ${\displaystyle M_{i+1}}$ ${\displaystyle M_{i}}$ ${\displaystyle M_{1},M_{2},\ldots ,M_{n}}$

Gabai proves the existence of grained manifold hierarchies under the following conditions.

Theorem : Let it be a tightly grained manifold that is not an atoroidal rational homology sphere . Then has a grained manifold hierarchy. ${\displaystyle M}$ ${\displaystyle M}$

If is tight, then is also tight with the exception of . ${\displaystyle M\setminus N(S)}$ ${\displaystyle M}$ ${\displaystyle M=D^{2}\times S^{1}}$

This often allows induction proofs to be carried out over the lengths of grained hierarchies.

Window dismantling

A disc decomposition (engl. Disk decomposition ) is a grained manifold hierarchy, wherein the decomposing step surfaces in each of the circular discs are.

Disk decompositions can be used to determine the minimum gender of the Seifert surface of a knot . If there is a disc with a scar , then a Seifert surface is of minimal gender. ${\displaystyle \Sigma }$ ${\displaystyle K\subset S^{3}}$ ${\displaystyle S^{3}\setminus N(\Sigma )}$ ${\displaystyle \partial _{\Sigma }\cap N(K)}$ ${\displaystyle \Sigma }$

Similarly, disk decompositions can be used to calculate the Thurston norm in any 3-manifold . If a surface is actually embedded and has a disk decomposition, then minimizes the gender in its homology class, i.e. calculates the Thurston norm. ${\displaystyle M}$ ${\displaystyle \sigma \subset M}$ ${\displaystyle M\setminus N(\Sigma )}$ ${\displaystyle \Sigma }$

Invariants

The invariants of grained manifolds include grained Floer homology, grained Khovanov homology, and grained topological quantum field theory.

literature

• David Gabai : Foliations and the topology of 3-manifolds. J. Differential Geom. 18 (1983) no. 3, 445-503. pdf
• Martin Scharlemann : Sutured manifolds and generalized Thurston norms. J. Differential Geom. 29 (1989) no. 3, 557-614. pdf
• Danny Calegari : Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007. ISBN 978-0-19-857008-0 (Chapter 5)

Web links

• Sutured Manifold (MathWorld)

Individual evidence

1. ↑ Theorem 4.2 in Gabai, op.cit.
2. ↑ Lemma 3.5 in Gabai, op.cit.
3. ↑ A. Juhász: Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6: 1429-1457 (2006).
4. ↑ E. Grigsby, Yi Ni: Sutured Khovanov homology distinguishes braids from other tangles. Math. Res. Lett. 21 (2014), no. 6, 1263-1275.
5. ↑ D.Matthews: Chord diagrams, contact-topological quantum field theory and contact categories. Algebr. Geom. Topol. 10 (2010), no. 4, 2091-2189.