I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.
Sutured guts: The following definition of sutured guts is cited from the preprint Guts in Sutured Decompositions and the Thurston Norm by Agol and Zhang. Let $(M,\gamma)$ be a taut sutured 3-manifold, where $\gamma$ is a collection of disjoint annuli and tori on the boundary $\partial M$. Let $R(\gamma)=\partial M\setminus int(\gamma)=R_+\cup R_-$, where $R_+$ and $R_-$ are oriented according to the orientation of $\gamma$. We further define
- A product disk is a disk $D$ in $M$ such that $D\cap \gamma$ consists of two arcs.
- A product annulus is an essential annulus $A$ in $M$ such that one component of $A$ lies on $R_-$ and another component of $R_+$ lies on $R_-$.
Decompose $(M,\gamma)$ along a maximal collection of disjoint nonparallel product annuli and product disks, throw away all product components (called the window) of the resulting sutured manifold, the remaining sutured manifold $(\overline M,\overline \gamma)$ is called the sutured guts of $(M,\gamma)$. It is well defined up to isotopy by JSJ-decomposition.
Pared guts: In this artical The minimal volume orientable hyperbolic 2-cusped 3-manifold by Agol, the guts of a pared manifold $(M,P)$ is defined as follows. A pared manifold $(M,P)$ is a pair where $M$ is a compact orientable irreducible 3-manifold and $P$ is a union of essential annuli and tori on the boundary $\partial M$, such that every abelian noncyclic subgroup of $\pi_1(M)$ is peripheral with respect to $P$, and every mapping of an annulus $(A,\partial A)\rightarrow (M,P)$ that is injective on the fundamental group can be homotoped into $P$.
Let $(M,P)$ be a pared manifold such that $\partial_0M:=\partial M\setminus P$ is incompressible. There is a canonical family of annuli (I suspect that one should also consider tori here?) $(A,\partial A)\subset (M,\partial_0M)$ which is the maximal collection of nonparallel essential annuli such that every other essential annulus $(B,\partial B)\subset (M,\partial_0M)$ can be relatively isotoped to be disjoint with $A$. A components of $M\setminus A$ is called a window if it is an ($I$-bundle, $\partial I$-bundle). Let $W$ be the union of such components. Then $M\setminus W$ is called the pared guts of $(M,P)$, which is also a pared manifold, with pared locus $\partial(M\setminus W)-\partial_0M$.
My question is:
- How to apply the JSJ-decomposition to see the well definedness of the sutured guts? I thought about the Characteristic Pair Theorem to produce a canonical seifert pair $(\Sigma,\Phi)\subset (M,R(\gamma))$, the frontier of the $I$-bundle components of $(\Sigma, \Phi)\subset (M,R(\gamma))$ seems to be the desired maximal collection of product annuli and product disks, but I am not sure. I also found that in the paper The virtual fibering theorem for 3-manifolds by Friedl and Kitayama, before Lemma 3.2 they said the guts may depend on the choice of the disks and annuli, which confuses me.
- Why is the pared guts well defined? In other words, why is there a canonical set of annuli as in the definition? Is it from the Characteristic Pair Theorem?
- What is the relation between pared guts and the sutured guts? Why do we need to consider product disks in the sutured case, but not in the pared case?
