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

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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)?

This post imported from StackExchange MathOverflow at 2014-10-25 10:37 (UTC), posted by SE-user Daniel Moskovich

asked Feb 16, 2014
retagged Oct 31, 2014

"It doesn't seem to be related to physics" ... wrong! Braids, knots, etc. are some of the fundamental tools using which many concepts in modern condensed matter and high energy physics can be adequately expressed. Yes, it would be nice to have a physics-related motivation for the question, but given the growing significance of braids (and not because of my own personal bias) for modern physics, such questions should, IMO, be considered useful for a physics audience.

I think that you would be right if everybody was importing such questions without any obvious physics reference. Despite that, we can be flexible sometimes and maybe ask the user to state the relevance to physics (maybe)? In any case, the particular question is not unrelevant.

Maybe in the mid-term exam ;). Actually, this is what I meant, that it should be the user's who is making the question job to give the physical background/motivation. In the meanwhile, and since I am not* an expert on the subject let me suggest Baez's work on higher dimensional Lie algebras and the relation of braidings with the scattering amplitudes in 2+1 dimensions. I have made some relevant question (not on the braids but related to them) before but got no answer. Especially after such a discussion maybe Daniel should do so.

@conformal_gk, but it's not imported by the author, Daniel probably is not aware that this post is here. Although I know Daniel in person, it would feel a bit weird if I ask him to "defend" a post that he never intended to put here.

Yiyang invited me to join Physics Overflow... so now I should comment here.

Much quantum topology for knots and links (e.g. the Jones polynomial) factors through Markov's Theorem, which relates the theory of braid groups (essentially algebra) with the theory of knots and links (low dimensional topology) by providing necessary and sufficient conditions for two different braids to have closures which represent ambient isotopic topological objects in 3 dimensions. Braid groups live in a world with much less structure than low dimensional topology.

Essentially, I'm specifically asking whether the injection of braids into string links leads to speedups (thus whether it is also an "injection of complexities"), in the way that injecting integers into real numbers leads to speedup proofs. I'm trying to understand an aspect of in what sense the sequence "first this crossing, then that crossing" that we have in a braid, and that occurs in the statistical mechanics context (or in the Witten TQFT context) from which the Jones polynomial arises is fundamental. Hiding behind this is the vague question of how fundamental the temporal order of things in TQFT (the "time axis", "Morse function", or whatever) really is, and whether there is something concrete to be gained by dropping it.

My vague intuition is that the answer to the above question is probably "no", but that there is a close variant for which the answer is "yes".
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