Abstract
We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surgery yields a 3-manifold with an essential lamination. Hence such a knot has property P and satisfy the cabling conjecture.
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