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Braid group can be viewed as a symmetry group with a "one more dimension to pass through". Is there any "Galois theory", where the braid groups plays analoguos role as a symmetry groups in a native Galois theory? Can this be related to the purposes of low-dimensional topology?

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This question strikes me as a little vague. Can you be more precise about what you're looking for and what you might want such a notion to do? –  Qiaochu Yuan Jun 19 '12 at 23:19
    
Construct an object like algebraic equation and a system of "roots", which symmetry can be described by a braid group. More preciesly - I'am thinking about a new machinery that can be used to describe knots naturally, some "equations" related to knots and some "groups of permutations of roots" which preserves isotopy. –  Andrew Jun 19 '12 at 23:45
    
It's not clear to me what you're asking exactly, but there are well-known analogies between the knot group (the fundamental group of a knot complement) and certain Galois groups. See for instance arxiv.org/abs/0904.3399. I'm not aware of Galois-theoretic analogues of the braid group though. –  Hiro Lee Tanaka Jun 20 '12 at 0:24
    
Have a look at this paper: front.math.ucdavis.edu/1204.4242 –  Ian Agol Jun 20 '12 at 0:47
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The braid groups are fundamental groups (of configuration spaces of points in the plane) so we have a standard Galois correspondence. –  Bruce Westbury Jun 20 '12 at 10:38