Accidental surfaces in knot complements

Abstract

It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot exterior. In this paper, we investigate closed incompressible surfaces in knot complements with this property. We show that if a closed, incompressible, non-boundary-parallel surface in a knot complement has such loops, then they determine the unique slope on the boundary of the knot exterior. Moreover, if the slope is non-meridional, then such loops are mutually isotopic in the surface. As an application, a necessary and sufficient condition for knots to bound totally knotted Seifert surfaces is given.

Non-Triviality of Generalized Alternating Knots

“…It follows from Lemma 3.3 that r(F, K) ≥ 1 if and only if F ∩ E(K) is incompressible in E(K), and r(F, K) ≥ 2 if and only if F ∩ E(K) is incompressible and ∂-incompressible in E(K), where E(K) denotes the exterior of K in S 3 . The following theorem is a mild generalization of Corollary 2 in [4]. Theorem 2.1.…”

Totally knotted Seifert surfaces with accidental peripherals

“…First, we will construct a knot K on the boundary of a handlebody V in S 3 such that K is separating in ∂V , ∂V − K is incompressible in V , and K bounds a genus one Seifert surface in V . Next, re-embedding (V, K) into S 3 so that ∂V becomes to be incompressible in S 3 −intV , we obtain a totally knotted Seifert surface of genus n > 1 as the closure of a component of ∂V − K. This follows a version of a result in [5]. Proof of Theorem 3.1.…”

“…It is known ( [1]) that an accidental slope of a closed incompressible surface is an integer or 1/0, and it was shown that F has a unique accidental slope. Furthermore if the accidental slope is integral, its accidental annulus is unique up to isotopy ( [5] In [5], it is conjectured that the integral accidental slope is unique for all accidental incompressible closed surfaces in E(K). By Theorem 1.3, if this conjecture is true, we can conclude that the integral accidental boundary slope is unique.…”

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

“…S 3 , and showed in [5] some classes of knots deny the Menasco-Reid conjecture [11], the conjecture that hyperbolic knot complements in S 3 do not contain closed totally geodesic embedded surfaces. In [6], several applications to Dehn surgery on hyperbolic knots were given.…”

“…It is known that the slope of a properly embedded surface with accidental peripherals in the exterior of a knot in S 3 is integral or ∞ [14, Theorem 1.4]. In [5] and [6], Ichihara and Ozawa described several properties of accidental closed essential surfaces embedded in knot complements in…”