On the Stable Equivalence of Plat Representations of Knots and Links

We are interested in the question of the decideability of the classical knot problem. A knot is the embedded image of a circle S 1 in Euclidean 3-space E 3 . If L 1 , L 2 are knots, then L 1 ≈ L 2 if there is an orientation-preserving homeomorphism h: E 3 ⟶ E 3 with h(L 1 ) = L 2 . By the “knot problem” we mean: given two arbitrary tame knots L 1 , L 2 (a knot is tame if it is equivalent to a polygonal knot), decide in a finite number of steps whether L 1 ≈ L 2 . The object of this paper is to show that the knot problem is ‘'stably equivalent“ to a problem of deciding membership in the double cosets of a distinguished subgroup K 2n of the classical braid group B 2n [1].