I still do not see what does this has to do with anything !

If we have a TIB tangle,

*any* TIB tangle ( that is, a knot topologically equivalent to the unknot, a concealed tangled straight line...), and we connect the two free ends, we get a tangled closed "loop", topologically equivalent to a ring. Now, if we cut this tangled closed loop

*anywhere ( *i.e., cut a bight somewhere, so we get two new ends - any bight, anywhere in the tangle, not only the bight that happened to be the eye of the eye-knot ), and we hold those two new ends, so they do not get entangled within the remaining tangle of which they are now the new ends, this remaining tangle would be topologically equivalent to the unknot, of course...If we connect again those two new ends, we get a tangle again topologically equivalent to a ring, and if we cut again this tangled ring / closed loop, anywhere, we get again two new ends, and a tangled straight line between them, etc...Perhaps I am missing something obvious...but what I see seems also obvious !

Cutting a topologically equivalent to a ring "closed" tangle, we get a topologically equivalent to a straight line "open" tangle - and if we connect the ends of this later, we get something as the former, etc.