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In this paper we provide a solution to the double coset problem for the braidgroup $B_n$ modulo the Hilden subgroup $H_n.$ This result demonstrates that, asin the case of braid closures, the Link Problem for plat closures is "stablyequivalent" to a solvable algebraic problem. A particularly interesting featureof the proof is the, like Garside's solutions to the Word and ConjugacyProblems, it too relies on Garside's decomposition of braids in $B_n.$

The Hilden Double Coset Problem in Braid Groups