Balanced taut sutured manifolds and characteristic product regions

A sutured manifold $(M,\gamma )$ consists of a compact oriented $3$-manifold $M$ with boundary and a union $\gamma$ of pairwise disjoint annuli and tori in $\mathrm{\partial }M$. Each annulus contains a suture, its oriented core circle; write $A(\gamma )$ for the annuli and $s(\gamma )$ for the union of their cores. Orient each component of $R(\gamma )=\mathrm{\partial }M\setminus \mathrm{int}(\gamma )$ so that its boundary orientation agrees in the annulus with the homology class of the suture. The regions whose normal points out of $M$ form $R_{+}(\gamma )$ and those whose normal points into $M$ form $R_{-}(\gamma )$. The sutured manifold is balanced when $M$ has no closed component, $\chi (R_{+})=\chi (R_{-})$, and every component of $\mathrm{\partial }M$ meets an annular component of $\gamma$. It is irreducible when every embedded $2$-sphere in $M$ bounds a $3$-ball in $M$. A properly embedded surface $S$ is incompressible when it has no compressing disk. A compressing disk is an embedded disk $D\subset M$ with $\mathrm{\partial }D\subset S$, $D\cap S=\mathrm{\partial }D$, and $\mathrm{\partial }D$ an essential simple closed curve in $S$ (that is, it does not bound a disk in $S$). For a compact surface $S$ put $x(S)={\sum }_{S_j}\max (0,-\chi (S_j))$, where the sum is over its connected components. The Thurston norm $x(\alpha )$ of a relative homology class $\alpha \in H_2(M,\gamma ;\mathbb{Z})$ is the minimum of $x(S)$ over properly embedded oriented representatives $S$ with $\mathrm{\partial }S\subset \gamma$ and $[S]=\alpha$. Such a representative is norm-minimizing when it is incompressible and $x(S)=x(\alpha )$. A balanced sutured manifold is taut when it is irreducible and $R(\gamma )$ is norm-minimizing in $H_2(M,\gamma ;\mathbb{Z})$.

A product sutured manifold is $(R\times [0,1],\mathrm{\partial }R\times [0,1])$ for a compact oriented surface $R$, with positive and negative regions $R\times \{1\}$ and $R\times \{0\}$. A properly embedded oriented surface $S\subset M$ is a decomposing surface when every component $\lambda$ of $S\cap \gamma$ is either a properly embedded nonseparating arc in an annular component meeting its suture once, a simple closed curve in an annular component homologous to that suture, or a homotopically nontrivial curve in a toroidal component, with all such curves in one torus component representing the same homology class. A product disk is a decomposing surface that is a disk and meets the sutures in exactly two points. A product annulus is a decomposing annulus with one boundary circle in $R_{+}$ and one in $R_{-}$. A horizontal surface $S$ in a balanced sutured manifold is a decomposing surface with no closed component, $\mathrm{\mid }\mathrm{\partial }S\mathrm{\mid }=\mathrm{\mid }s(\gamma )\mathrm{\mid }$, $[S]=[R_{+}]$ in $H_2(M,\gamma ;\mathbb{Z})$, and $\chi (S)=\chi (R_{+})$. The sutured manifold is vertically prime when every horizontal surface is parallel to $R_{+}$ or $R_{-}$, where parallel means that the surfaces cobound an embedded product between them.

When $(M,\gamma )$ is irreducible, $\gamma$ has no torus component, and $R_{+}$ and $R_{-}$ are incompressible and diffeomorphic, a product region is a submanifold $\mathrm{\Phi }\times [0,1]\subset M$ for a compact possibly disconnected surface $\mathrm{\Phi }$ such that $\mathrm{\Phi }\times \{0\}$ and $\mathrm{\Phi }\times \{1\}$ are incompressible subsurfaces of $R_{-}$ and $R_{+}$, respectively. A characteristic product region is a product region $E\times [0,1]$ such that every product region $\mathrm{\Phi }\times [0,1]$ can be carried into it by an ambient isotopy of $M$.

Ni, Knot Floer homology detects fibred knots, §§2.1 and 6.1

Juhasz, Floer homology and surface decompositions, §§2 and 9