Here is a paper that, I think, underlines some of the difficulties in recognising the unknot using the physical process of "pulling tight".
Nontrivial embeddings of polygonal intervals and unknots in 3-space
by Cantarella and Johnston. They prove that there are "stuck" stick unknots for any stick number $n \geq 6$. That is, polygonal knots that are topologically the unknot but cannot be unknotted without changing the lengths of the sticks (or adding breakpoints).
As another piece of negative evidence we have the paper
by Coward and Hass. They give a link (made of "rope" - thus each component has a given thickness and length) which is topologically unlinked but cannot be physically unlinked.
