The Hilden Double Coset Problem in Braid Groups

In this paper we provide a solution to the double coset problem for the braid group $B_n$ modulo the Hilden subgroup $H_n\mathrm{.}$ This result demonstrates that, as in the case of braid closures, the Link Problem for plat closures is “stably equivalent” to a solvable algebraic problem. A particularly interesting feature of the proof is that, like Garside’s solutions to the Word and Conjugacy Problems, it too relies on Garside’s decomposition of braids in $B_n\mathrm{.}$