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Thursday, April 20, 2006

**Abstract:** Given a tree braid group $B_nT$ on $n = 4$ strands, we are able to reconstruct the tree $T$. Thus tree braid groups $B_4T$ and trees $T$ (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1-dimensionality of trees. We use this bijection to solve a version of the isomorphism problem for tree braid groups with $n = 4$ strands. We also make some comments on the possibility of generalizing this solution to tree braid groups with more strands.