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If you have an invisible creature totally surrounded on all sides (and know for practical reasons it can't escape), like this:

enter image description here

… are all those surrounding it flanking it?

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    \$\begingroup\$ Total concealment having the effects it does means a good answer might help answer this question. \$\endgroup\$ – Hey I Can Chan Jul 20 '16 at 0:25
  • \$\begingroup\$ Cover seems to be different than total concealment. Solid Snake uses a cardboard box as concealment. He's "covered" by the box (tho people still can attack it if he's still there and hit him). Maybe I better draw a picture and upload it... \$\endgroup\$ – Jesse Cohoon Jul 20 '16 at 0:35
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    \$\begingroup\$ Invisibility is a lot like total concealment, though, which says, "You can’t execute an attack of opportunity against an opponent with total concealment, even if you know what square or squares the opponent occupies." \$\endgroup\$ – Hey I Can Chan Jul 20 '16 at 0:41
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    \$\begingroup\$ @HeyICanChan so surrounding it gives no mechanical advantage other than "hey, we know this thing is here but we have it surrounded so it can't escape" :-P \$\endgroup\$ – Jesse Cohoon Jul 20 '16 at 0:44
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    \$\begingroup\$ I don't have an answer (and don't plan on having one), but I think it's closer to, "We think this thing is here, but we have no idea what it's doing!" (As an aside, I really try to avoid such situations in campaigns I DM after researching fighting in darkness—it's just too complicated.) \$\endgroup\$ – Hey I Can Chan Jul 20 '16 at 1:01
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Yes, you can flank an invisible creature.

The confusion appears to be due to conflating the ability to take attacks of opportunity against a creature and threatening a creature's square. These are not the same thing, and must be understood as distinct and separate.

Threatened Squares

You threaten all squares into which you can make a melee attack, even when it is not your action. … An enemy that takes certain actions while in a threatened square provokes an attack of opportunity from you.

Notice that the attack of opportunity depends on threatening the square, but threatening the square does not depend on being able to make the attack of opportunity. You still threaten squares even when you can't take an attack of opportunity into them. If some other reason causes you to be unable to take an attack of opportunity into a square, you still threaten the square. So, you still threaten the square of an invisible creature in your reach.

Since flanking requires only that an ally threatens the square

Only a creature or character that threatens the defender can help an attacker get a flanking bonus.

… the fact that invisible creatures are proof against attacks of opportunity (due to having total concealment) is irrelevant to flanking — the ally flanker still threatens the square the first flanker wishes to attack into, therefore they will get the bonus for flanking (assuming they pass the 50% miss chance and can roll the attack at all).

This is true even without surrounding the invisible creature. Surrounding it only makes it easier to reliably locate the correct square to flank and attack into, but makes no difference to whether the square being attacked into threatened — creatures threaten squares regardless of whether anything is there to attack. The usual means of determining which square an invisible opponent is in are equally effective for locating the creature for the purpose of positioning for a flanking bonus.

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I would say yes:

My thoughts: Flanking is the result of a tactical disadvantage for a defender. I know of no restrictions on flanking based upon the status of an attacker (other then the ones mentioned, obviously). The primary advantage for a flanking attacker is that the defender is less able to anticipate the attacks (threatened).., not the other way around. (FYI: A sneak attack versus an invisible creature is not possible, the target has to be seen.)

Note: Without pinning/tripping/etc an adversary, there's always a chance of escape. On its' turn, the invisible creature may, for example, try to:

  1. Bull-rush a flanking creature (not provoking any attacks of opportunity, for having total concealment), OR;
  2. Tumble (DC 25, trained only) his way out. If tumbling is successful, the invisible creature moves through the occupied area at half speed WITHOUT provoking any attacks of opportunity. If the invisible creature fails the tumble-check, it can't move through the occupied area but still doesn't provoke any attacks of opportunity for having total concealment (according to RAW), OR;
  3. Grapple, receiving a +2 on the initial grab (melee touch attack), penalties to the defenders AC apply as well. Etc.

RAW:

Flanking depends on being threatened, which only mentions the ability to attack a square, it does not rely on being able to execute any attacks of opportunity, or targets, or anything else AFAIK. A square is an area in my book, not an opponent. This eliminates concealment as a factor (for the area a totally concealed occupies can still be attacked, thus is threatened), just like the differences between line of sight and line of effect, for example.

flank:

To be directly on the other side of a character who is being threatened by another character. A flanking attacker gains a +2 flanking bonus on attack rolls against the defender. A rogue can sneak attack a defender that she is flanking.

threaten:

To be able to attack in melee without moving from your current space. A creature typically threatens all squares within its natural reach, even when it is not its turn to take an action. For a Medium or Small creature this usually includes all squares adjacent to its space. Larger creatures threaten more squares, while smaller creatures may not threaten any squares except their own.

My only concern is the "typically" & "usually," I interpret this as, unless mentioned otherwise (f.e. in case of reach attacks that don't threaten adjacent squares)

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    \$\begingroup\$ Cite for tumble working even if you fail the check? Normally, if you try to tumble though an enemy and fail the check, you both abort the move and provoke an AOO. Invisibility protects from the AOO, but I don't see why it would allow you to keep moving. \$\endgroup\$ – topquark Jul 20 '16 at 8:05
  • \$\begingroup\$ @topquark: I stand corrected. \$\endgroup\$ – agodwithoutananswer Jul 20 '16 at 8:59
  • \$\begingroup\$ It actually raises a question: -> Do occupied squares increase Tumble DCs for an invisible creature? \$\endgroup\$ – agodwithoutananswer Jul 20 '16 at 9:54
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No

Threatened Squares

You threaten all squares into which you can make a melee attack...

When it's not your turn, because the opponent is invisible, and thus has total concealment, and thus you are prevented from attacking, you don't threaten because you can't "make a melee attack".

To threaten is not a static state, but defined by your current situation and with respect to a given opponent; you must be armed, be able to reach the opponent, the opponent cannot have cover or be totally concealed, and you must not otherwise be prevented from attacking. You must be able to attack at the time whether-or-not-you-threaten comes into question (when an enemy provokes, or an ally attacks from a flanking position).

To "threaten" and being able to make an attack of opportunity are distinct, but there is no subset of consequence where you can attack, but can't make an attack of opportunity. Effectively, if prevented for any reason from making an attack of opportunity, you only threaten on your turn, and it never matters on your turn.

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