Normal surfaces in knot complements

We extend normal surface Q-theory developed in compact triangulated 3-manifolds to some non-compact 3-manifolds and apply the Q-theory to knot complements. We also give an algorithm to find a normal surface representing a minimal Seifert surface of a non-fibered knot in the knot complement. The figure-eight knot is presented as a fibered knot which does not have any either normal or almost normal representation of a minimal Seifert surface of the knot in its complement in S 3.