The classification of nonsimple algebraic tangles (Q1910169)

A tangle \(T\) is a pair of strings \(t_1 \cup t_2\) properly embedded in a 3-ball \(B\). It is called simple if the complement of an open regular neighborhood of the strings is a simple 3-manifold, i.e. a manifold which is irreducible, \(\partial\)-irreducible, atoroidal, and anannular, or equivalently (by Thurston), a manifold which admits a hyperbolic metric with totally geodesic boundary. A tangle is algebraic in the sense of Conway if it can be obtained by summing rational tangles together in various ways. In general it is difficult to determine whether a tangle is simple. The main theorem of this paper will give a complete classification of all nonsimple algebraic tangles. In particular, if \(T\) is a Montesinos tangle \(M(r_1,r_2)\), then it is nonsimple if and only if \(T = M(r_1,1/2n)\) or \(M(1/p,1/q)\). For Montesinos tangle \(T = M(r_1,\dots, r_n)\) with \(n \geq 3\), it is nonsimple if and only if, for some integer \(k\), either \(r_1\) or \(r_n\) is \(1/2k \bmod Z\). We will show that the simplicity of the tangles given by Jaco, Ruberman and Thurston follows immediately from our result.