A SURVEY ON ACCIDENTAL SURFACES IN KNOT COMPLEMENTS

Let K be a knot in the 3-sphere S3, and F a properly embedded surface in the exterior E(K) of K in S3. A non-trivial loop l in F is called an accidental peripheral if it is freely homotopic into ∂E(K) in E(K) but not in F. Here, an annulus A connecting l and a loop l ′ in ∂E(K) is called an accidental annulus for l. We define an accidental surface as such surface F with an accidental peripheral. We always assume that surfaces are essential, that is, incompressible, ∂-incompressible and not boundary-parallel. For a two-sided surface F in a 3-manifold M, this means that the induced homomorphism i # : π1(F) → π1(M) and the induced map i #: π1(F, ∂F) → π1(M,∂M) are injective. The existence of an accidental peripheral causes that i#(π1(F)) contains an element which is conjugate to some element of the peripheral subgroup π1(∂M). Thus, in the case that M is hyperbolic, ρ(i#(π1(F))) contains an accidental parabolic element, where ρ: π1(M) → PSL2(C) is a faithful discrete representation. 2. Preparations Lemma 2.1 ([6]). Let F be an accidental essential two-sided surface properly em-bedded in E(K). Then there exists an embedded accidental annulus for F. 3. Accidental closed surfaces In this section, we treat with accidental closed surfaces. Let S be an accidental closed surface in E(K). Theorem 3.1 ([3]). All accidental annuli A determine the unique slope A∩∂E(K). Moreover, if those slopes are non-meridional, then all accidental annuli are mutually isotopic rel.S ∪ ∂E(K). Thus, we can define the accidental slope of S as a loop A∩∂E(K) for an accidental annulus A. Theorem 3.2 ([1]). Any accidental slope is meridional or integral. Thus, there is a meridionaly compressing disk for S or K is isotopic onto S, according to the accidental slope is meridional or integral. A properly embedded surface in E(K) with non-empty boundary is said to be strongly essential if F is essential and at least one component of E(K) − intN (F;E(K)) is ∂-irreducible. Theorem 3.3 ([3]). The following are equivalent