Any normal means by which you'd open a door or window or break an object. It's just harder to do so (though just picking the lock may not be enough).
The last line of this spell handles this
While affected by this spell, the object is more difficult to break or force open; the DC to break it or pick any locks on it increases by 10. (PHB 215)
So if the DC to force open a locked window or door is normally 10, then one under the effect of Arcane Lock is now 20 (and so on and so forth).
Basically, the effect of this spell is to make an easy check a hard one, a medium check a very hard one, and a hard check nearly impossible.
As far as the text "impassible", it seems to mean in this context that the door/window can't be opened by normal means except with the higher DC. It's unclear to me whether or not picking a lock has any effect on this text, though it seems quite clear that forcing the door open is in bounds. I would definitely rule that at the very least picking the lock(s) on the door would ease the DC to force it open (As you're only dealing with the magic and not any physical locks).
To determine whether these things are valid targets of the spell, they must be included in this portion of the spell description:
The object can be a door, a box, a chest, a set of manacles, a padlock, or another object that contains a mundane or magical means that prevents access.
A target that is held shut by a mundane lock or that is stuck or barred becomes unlocked, unstuck, or unbarred.
Notably, "another object that contains a mundane or magical means that prevents access" will be how we determine special cases.
So,
- knotted rope tying up a PC
Yes. I would argue that a knot that is tying someone's hands together is functionally similar to a set of manacles and the knot could be considered a mundane means of preventing access. So long as there is a single knot because:
If the object has multiple locks only one of them is unlocked.
Locks in this case could extend to knots but you could leave that up to DM discretion.
- continuous chain without a lock that is holding a chest shut
Yes. This chain is a means of preventing access to the chest. Whether the chain magically falls to the ground or is unlinked without actually breaking the link could be up to DM discretion, but I tend to agree with the other comments suggesting that the spell doesn't actually break an object.
- buckle holding a belt together
Yes. My first thought on this one was no, but technically the belt prevents access into the trousers. I would argue that the prong on a traditional buckle is close enough to a bar on a door to be a valid target for the spell. To what extent the belt is undone is up to DM discretion.
- vine/tentacle wrapped around something (either an entry or a PC limb) (considering that manacles are a valid knock target)
Let's break this one up...
For an entry, Yes. If the vines are sufficiently obstructing a doorway to prevent access they could be considered in the same way a barred door prevents access.
For the PC's limbs, No. The reason this is different from manacles or the knotted rope is that there is no mechanism that is actually locked, barred, stuck, or in the rope's case knotted. The reason I said the chain would be valid and this would not is because for the chain to form a loop the last link made would have to be closed essentially sealing the loop. The vine or tentacle doesn't lock or tie itself, and it doesn't produce an enclosed loop.
- Fingers grasping an object
No. The target must be an object, not a creature
- heavy hatch covering an escape hole
No. I think there's an important distinction to be made here about the result of casting this spell. Nowhere in the description does it indicate that the target that is being unlocked, unstuck, or unbarred opens. A chest with a locking mechanism will become unlocked and many DMs in this case might describe the chest swinging open as a result, but RAW there is no indication that this is an inherent feature of the spell. The spell description states :
A target that is held shut by a mundane lock or that is stuck or barred becomes unlocked, unstuck, or unbarred.
And
If you choose a target that is held shut with arcane lock, that spell is suppressed for 10 minutes, during which time the target can be opened and shut normally.
Of course for a mechanical lock such as a padlock, it is usually opened due to being unlocked, but the door or chest it is preventing access to wouldn't necessarily swing open as a result.
The spell won't make the hatch swing open, and it also wouldn't make the hatch any lighter.
- glue/resin sealing a window
Yes. It's a mundane means of preventing access. Knock would definitely unstick the window.
As for the socks, you're out of luck. Unless you tie someone's hands together with them, but in that case remember that Knock would only undo one knot.
Sorry for the wall. I'm open to discuss these if you disagree.
Best Answer
Up to your DM
This is one of those edge cases where the DM needs to adjudicate.
However, whether it should work for a book with a lock (like a modern diary, as you pointed out), or a book without such a locking mechanism, going by the examples the spell does give, I'm not sure what I would rule on this.
The spell says (PHB, p. 215):
At least most of these are things that have locks on them, or at least can have locks on them. A book can have a lock on it (i.e. the locked diary), so then the question becomes "does arcane lock lock things that can have locks on them, but don't?"
The spell also says:
It says "to break it or pick any locks on it". This, to me, suggests that the objects should have a lock, since this seems to have been worded with a locked door in mind. Those are the two main ways to get past a locked door. Under this interpretation, a DM could rule that you can cast arcane lock on a book with a lock already on it, but not a book without a lock.
However, the fact that it says "or", could equally be taken to mean "if it has a lock, you can pick the lock, but if not, you'd have to break it", meaning that the door, window, etc, doesn't need to have a lock on it to be a valid target of arcane lock. But then, what stops this spell from targeting any object? Already we have an inconsistent list with "chest, or other entryway"...
So we're still left with "up to the DM". If the DM did allow it, then if someone wants to open your arcane locked book, they'd need to either pick the lock, or break it open (with great difficulty) to see the pages inside. This would, of course, be the same for any such item the DM ruled would be valid (backpacks, etc).