High distance Heegaard splittings from involutions

Fixed an oriented handlebody H = H + with boundary F, let η ( H + ) = H − be the mirror image of H + along F, so η ( F ) is the boundary of H − , for a map f : F → F , we have a 3-manifold by gluing H + and H − along F with attaching map f, and denote it by M f = H + ∪ f : F → F H − . In this note, we show that there are involutions f : F → F which are also reducible, such that M f have arbitrarily high Heegaard distances.