This paper gives the isotopy classification of incompressible and \(\partial\)-incompressible surfaces (orientable or not) in the complement of a 2-bridge knot \(K_{p/q}\). The only closed incompressible surfaces are boundary parallel tori. For each continued fraction expansion \(p/q=r+[b_ 1,...,b_ k]\) there is a branched surface \(\Sigma (b_ 1,...,b_ k)\) in \(S^ 3\) which contains all the surfaces obtained by plumbing together k bands in a row, the ith band having \(b_ i\) half- twists. (These surfaces all have boundary \(K_{p/q}.)\) All the incompressible \(\partial\)-incompressible surfaces are obtained by plumbing ”multiple bands”.
The questions of which of these are connected and which are orientable are discussed, but not dealt with completely. The slopes of their boundary curves are given. The orientable incompressible Seifert surfaces for K all have the same genus, and are all isotopic if and only if in the unique continued fraction expansion of p/q with all \(b_ i's\) even at most one is not \(\pm 2.\)
Except when \(K_{p/q}\) is a torus knot, Dehn surgery always gives an irreducible 3-manifold, and almost all such are hyperbolic but not Haken.