Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

Abstract

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into T × I (T is a torus with one boundary component). We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto [12] and Baker [2], [3].

The subgroup of the Goeritz group of the Heegaard splitting induced by an openbook decomposition consisting of elements preserving the binding

“…In this subsection, we recall some definitions about simple closed curves on a closed surface of genus two and GOF-knots on the boundary of a handlebody of genus two, which is appeared in [6], [7], [20], and will be used for the proof of Proposition 6.1. All curves in a surface is assumed that they intersect each other minimally and transversely.…”

“…Fact 6.1. ( [20]) For each i = 1, 2, the number of arcs in a diagram of l in P (Σ; l 1 , l 2 ) starting from and ending on l i + is the same as that of arcs in a diagram of l in P (Σ; l 1 , l 2 ) starting from and ending on l…”

“…Let H be a handlebody of genus two, oriented as a subset of R 3 . In [20], an oriented simple closed curve l in ∂H is called a GOF-knot of H if there is an homeomorphism between Σ 1,1 × [0, 1], ∂Σ 1,1 × { 1 2 } and (H, l) as pair of unoriented manifolds. Note that if l is a GOF-knot of H, then l with the opposite orientation is also a GOF-knot of H.…”

Reidemeister torsion of a 3-manifold obtained by an integral Dehn-surgery along the figure-eight knot

“…(1.1.1) If {−1, 0} ⊂ {p, q, r}, then T = φ and the entire manifold is S 2 × S 1 , (1.1.2) if {p, q, r} = {−1, ǫ, ǫ} or {0, ǫ − 1, ǫ − 1} for ǫ ∈ {±1}, then T = φ and the entire manifold is a closed torus bundle over S 1 whose monodromy is periodic of order six, (1.1.3) if {p, q, r} = {−1, ǫ, k} or {0, ǫ − 1, k − 1} for ǫ ∈ {±1} and some integer k and not in the cases above, then |T | = 1 and the decomposed piece is a Seifert manifold whose base surface is an annulus with one exceptional point of order |k|, Some of the cases in Theorem 1.1 has already known. For example, the cases (1.1.3) and (1.1.4), where the corresponding pretzel knot is so called "double twist knot" are stated in [22], the case (1.1.5) is stated in [6]. Our classification heavily depends on the result of [15], which classifies the exceptional fillings of the minimally twisted five-chain link.…”