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A nontrivial knot that can
be drawn with only two relative maxima in the vertical direction is called a 2-bridge
knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking,
a lamination in a manifold M is a foliation of M, except that it can have a
nonempty open complement in M, and very loosely speaking, the lamination is
essential if each leaf of it L is incompressible, i.e. inclusion of L into M
induces an injective homomorphism from π1(L) into π1(M). Our main result
is:
Theorem 2. Every 3-manifold obtained by surgery on a non-torus 2-bridge knot
admits essential laminations.
Some immediate corollaries are that these manifolds are covered by R3,
and have infinite fundamental group. So Property P is true for non-torus
2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or
a counterexample to the Poincare conjecture. We use general techniques
which are not specific to 2-bridge knots to find and explicitly construct these
laminations.
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