imagine two black holes approaching each other in a way that they will perform just very close flyby. You have a spaceship just below the event horizon of one black hole in line between centers of both black holes all the time. When the black holes are closest to each other you should find yourself outside of the event horizon for a moment as the influence of one black hole should cancel out some part of the influence of the another black hole ( in similar fashion as the influences cancel out in the exact middle of a planet ); therefore, enabling you to escape. However, current physics says that nothing can leave the event horizon. Where is the flaw in this scenario?
In short, it doesn't work.
the event horizon is defined as the region of space which you can't escape from, If you are between two black holes, and you can escape then by definition you are not inside the event horizon. In the case when there are two black holes, the black hole is not a Schwazchild black hole (which is a solution of Einstein's equations for a single point, uncharged non-spinning mass) and the event horizon is not spherical. To determine the shape of the event horizon for two masses you need to use numerical solutions of the equations, you can do this yourself with the Einstein Toolkit, but the computing requirements are very high. A supercomputer is required to get very far.
In the exact situation you draw, it is likely that the release of graviational radiation would lead to a merger rather than a pass but you would need to run the simulation to confirm this.
Now you can't just remain stationary inside the event horizon. If you started inside it you can't stay in the same place and wait for a the pass of the other black hole to pull you out. Inside the event horizon spacetime is flowing towards the singularity and even if the shape of the event horizon is changed by the passing of another black hole, nothing passes from inside to out.
The black holes will either merge or they won't. If they merge, they just become a bigger black hole, and we're back to square one.
If they don't merge, then...
When the black holes are closest to each other you should find yourself outside of the event horizon for a moment as the influence of one black hole should cancel out some part of the influence of the another black hole ( in similar fashion as the influences cancel out in the exact middle of a planet ); therefore, enabling you to escape.
There is no similarity there.
Most people are fixated on a phrase, often repeated in popular science, that "not even light can escape from a black hole", which leads people to believe that if you could somehow go faster than light, or if the bond with the black hole was a bit less strong, you could escape.
Well, you can't. That bond is not a finite amount. It's effectively infinite (emphasis on "effectively", because there's a trick here). Let me explain what I mean.
Inside the event horizon you're effectively as if in a different universe. That's a chunk of spacetime completely cut off from this universe. The reason why you can't escape is not because the pull is "too strong" or you can't move "fast enough". The real reason is that there is no path from inside the event horizon to the outside.
Pick any point inside the event horizon. Draw a straight line from there in any direction. Keep drawing. Eventually the line will end at the center of the black hole.
Okay, wrong direction. Pick another direction. Start drawing another straight line. Keep drawing. Ooops, hit the center again.
Back to the initial point and pick a completely different direction. Draw the straight line. Damn, hit the center again.
Start over, but now scribble some randomly-shaped, winding line. Holy cow, I hit the center again!
That's how it is. Spacetime inside the event horizon is closed into itself, will all possible trajectories ending at the center of the black hole. There is no possible escape route. No matter what, you will end up in the center.
Bringing in another black hole makes no difference, because the two black holes will merge, first their event horizons, then they will complete the merger. At any stage in this process, the same rule applies to their common event horizon: there is no path from inside that can break through to the outer universe.