Pairwise Incompressibility of Surfaces in Knot Complements

We study the properties of incompressible pairwise incompressible surfaces in knot exteriors. Let K be a prime almost alternating knot in S~3 and let F be an incompressible pairwise incompressible suface in S~3-K .Then there exist loops of type S~2 and of type PS~3 in F∩S~2_± .We prove that there are only finitely many such surfaces in S~3-K with n boundaries components for fixed n by discussing the properties of loops in F∩S~2_± ,and show that F is punctured sphere if K is a connected sum of two pretzel knots.