# Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.

This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, are related by ambient isotopy if and only if they are related by height-preserving ambient isotopy. This is a non-trivial fact whose proofs are all somewhat complicated (Stallings Theorem/ Magnus expansion/ embedding fibrations).

Given that there is no theoretical advantage to injecting braids into string links (distinct braids stay distinct), I wonder whether there is a computational advantage in doing so. Explicitly, given diagrams for $B_1$ and for $B_2$, is the minimum number of Reidemeister moves between them always realized for Reidemeister moves between braids? Or might the `shortest path between two braids' pass through string links?

Question: Is there an example of a pair of equivalent braid diagrams, considered as tangle diagrams, such that the minimum number of Reidemeister moves between them is increased if we allow only braid-like Reidemeister moves (i.e. if the result of each Reidemeister move must also be a braid)?
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