We study a canonical spanning surface obtained from a knot or link diagram, depending on a given Kauffman state. We give a sufficient condition for the surface to be essential. By using the essential surface, we can deduce the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class that includes all semiadequate, homogeneous knots and links, and most algebraic knots and links. In order to prove the main theorem, we extend Gabai’s Murasugi sum theorem to the case of nonorientable spanning surfaces.
