A presentation for Hilden’s subgroup of the braid group

Consider the unit ball, $B = D \times [0,1]$, containing $n$ unknotted arcs $a_1, \ldots, a_n$ such that the boundary of each $a_i$ lies in $D \times \{0\}$. We give a finite presentation for the mapping class group of $B$ fixing the arcs $\{a_1, \ldots, a_n\}$ setwise and fixing $D \times \{1\}$ pointwise. This presentation is calculated using the action of this group on a simply-connected complex.

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Published 1 January 2008