Closed incompressible surfaces in complements of wide knots and links

Abstract

We define the notion of wide knots (and links) and show that they contain closed incompressible nonboundary parallel surfaces in their complement. This is done by proving that these complements admit Heegaard splittings which are irreducible but weakly reducible, and using an extension of a result of Casson and Gordon. We then show that the class of wide knots and links is rather large, and that examples are easy to come by. We also show that the incompressible surfaces remain incompressible after most Dehn fillings.