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thesis, David M. Dahm was able to show that there is an injective homomorphism from this group into the automorphism group of the free group on n generators, so it is natural to identify the group with this subgroup of the automorphism group. One may also show that the loop braid group is isomorphic
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circles into the 3-disk. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy, and then we define a group operation on the equivalence
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loops is defined as the motion group of n disjoint circles embedded in a compact three-dimensional "box" diffeomorphic to the three-dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding
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loops are exchanges of two adjacent loops, and passing one adjacent loop through another. The topology forces these generators to satisfy some relations, which determine the group.
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with loop-like topologies within three dimensions of space and time.
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