Complements of hyperbolic knots of braid index four contain no closed embedded totally geodesic surfaces

We prove that if F is a closed essential surface embedded in the complement of a knot K of braid index four, then at least one of the following holds: (1) F is meridionally compressible, (2) K is isotopic to a simple closed curve on F, (3) there is an essential annulus properly embedded in the closure of the component of S3−N(F) which does not contain K. We obtain a corollary that there are no closed embedded totally geodesic surfaces in the complements of hyperbolic knots of braid index four.