Seifert surfaces for genus one hyperbolic knots in the 3–sphere

Abstract

The Kakimizu complex MS(K) for a knot K ⊂ S 3 is the simplicial complex with vertices the isotopy classes of minimal genus Seifert surfaces in the exterior of K and simplices any set of vertices with mutually disjoint representative surfaces. In this paper we determine the structure of the Kakimizu complex MS(K) of genus one hyperbolic knots K ⊂ S 3 . In contrast with the case of hyperbolic knots of higher genus, it is known that the dimension d of MS(K) is universally bounded by 4, and we show that MS(K) consists of a single d-simplex for d = 0, 4 and otherwise of at most two d-simplices which intersect in a common (d − 1)face. For the cases 1 ≤ d ≤ 3 we also construct infinitely many examples of such knots where MS(K) consists of two d-simplices.

On non almost-fibered knots

“…• γ is a power circle in H if γ represents a non-trivial power in π 1 (H), that is, if γ represents a power p ≥ 2 of some non-trivial element in π 1 (H). In the part 2 of Lemma 3.3 of [10], Valdez-Sánchez prove the next Lemma 1. Let γ ⊂ ∂H a circle which is non-trivial in H. Then γ has a companion annulus in H iff γ is a power circle in H; more precisely,…”

“…Proof. This follows from Lemma 3 and from the fact that there are no companion annuli on both sides of T i+1 with the same boundary slope (Lemma 5.1 of [10]).…”

“…, 4}. By Lemma 6.8 of [10], S is parallel to T i or T i+1 . Now, if S ∩ T = ∅, by the incompressibility of S and T i (i = 1, .…”

“…, 4) a power circle in H i cannot have companion annuli on both sides of T i . Then part 1 of Lemma 8.1 of [10] implies that H is hyperbolic. Lemma 3.…”

“…Companion annuli in genus two handlebodies. In this section much of the definitions and notations follow from L. Valdez-Sánchez [10].…”