Bigon criterion in dimension 3? The bigon criterion for surfaces says that if two simple closed curves $\alpha$ and $\beta$ embedded on a surface $\Sigma$ intersect in points $\{p_1,\dotsc,p_n\}$ and $\alpha$ and $\beta$ can be isotoped so that there are fewer points of intersection, then there is in fact a bigon on $\Sigma$ between $\alpha$ and $\beta$ (a bigon between $\alpha$ and $\beta$ is an embedded 2-dimensional disk $D$ with the boundary of $D$ partitioned into two arcs, one in $\alpha$ and the other in $\beta$, and otherwise $D$ is disjoint from $\alpha$ and $\beta$ — such a bigon then guides a simple to see isotopy that reduces the number of points of intersection between $\alpha$ and $\beta$).
I am interested in to what extent some generalizations of this hold in dimension 3.  The two possible cases I am interested in are:

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*We consider two surfaces $F_1$, $F_2$ embedded in a 3-manifold $M$.

Here a bigon would be a 3-dimensional embedded ball with interior disjoint from the surfaces and with boundary partitioned into two 2-dimensional disks — one in $F_1$ and the other in $F_2$.  Does the bigon criterion hold in this case?


*We consider a knot $K$ and a surface $F$ embedded in a 3-manifold $M$.

Here a bigon would be an embedded disk with interior disjoint from the knot and surface and with boundary partitioned into an arc in $K$ and an arc in $F$.  Does the bigon criterion hold in this case?
 A: EDIT: While looking for the reference I thought I remembered, I realised that the criterion I gave for surfaces was wrong.  I have attempted to fix it.

One version of the bigon criterion for surfaces in a three-manifold is as follows: Suppose that $M$ is a compact, connected, closed, oriented three-manifold.  Suppose that $F$ and $G$ are compact, closed, connected, two-sided, incompressible surfaces in $M$ which intersect.  Suppose that $F$ and $G$ are transverse.  So their intersection is a finite collection of curves.  Let $|F \cap G|$ be the number of these curves.
If $F$ and $G$ can be made disjoint (via isotopy) then there is a there is a product region in $M - (F \cup G)$.  Here a product region is homeomorphic to a surface $X$ crossed with an open interval so that $X \times \{0\}$ lies in $F$ and $X \times \{1\}$ lies in $G$ and $(\partial X) \times (0,1)$ is mapped to some components of $F \cap G$. [The "flying saucer" (three-ball) region you mention is a special case of a product region where $X$ is a disk.]
Thus, there is an isotopy making $F$ and $G$ disjoint that monotonically decreases $|F \cap G|$.

On the other hand, when isotoping a knot off of a surface, sometimes the intersection number must go up before it goes down.  So I do not see a version of the bigon criterion here.  (The bigon you describe, meeting the knot and the surface each only in an arc of its boundary, does appear in various places.  Google "thin position of knots in the three-sphere" for example.)
