Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries

Suppose [Formula: see text] and [Formula: see text] are disjoint simple closed curves in the boundary of a genus two handlebody [Formula: see text] such that [Formula: see text] (i.e. a 2-handle addition along [Formula: see text]) embeds in [Formula: see text] as the exterior of a hyperbolic knot [Formula: see text] (thus, [Formula: see text] is a tunnel-number-one knot), and [Formula: see text] is Seifert in [Formula: see text] (i.e. a 2-handle addition [Formula: see text] is a Seifert-fibered space) and not the meridian of [Formula: see text]. Then for a slope [Formula: see text] of [Formula: see text] represented by [Formula: see text], [Formula: see text]-Dehn surgery [Formula: see text] is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435–472.]. In this paper, we show that there exists a meridional curve [Formula: see text] of [Formula: see text] (or [Formula: see text]) in [Formula: see text] such that [Formula: see text] intersects [Formula: see text] transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery [Formula: see text] can arise from a primitive/Seifert position of [Formula: see text] with [Formula: see text] its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in [Formula: see text] is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.

ON TUNNEL NUMBER ONE KNOTS THAT ARE NOT (1, n)

We show that the bridge number of a tunnel number t knot in S3 with respect to an unknotted genus t surface is bounded below by a function of the distance of the Heegaard splitting induced by the t tunnels. It follows that for any natural number n, there is a tunnel number one knot in S3 that is not (1, n).

STABILIZING HEEGAARD SPLITTINGS OF COMPOSITE TUNNEL NUMBER TWO KNOTS

Suppose K # K′ is a composite tunnel number two knot where both K and K′ are tunnel number one knots. Let t1, t2 be unknotting tunnels of K and t′1 , t′2 be unknotting tunnels of K′ and assume that cl (S3 - N(K ∪ t1 ∪ t2)) and cl (S3 - N(K′ ∪t′1 ∪t′2)) are genus three handlebodies. Suppose {t1 , t′1} , {t2 , t′2} are tunnel systems of K # K′. Then we give some condition such that the two genus three Heegaard splittings induced by these tunnel systems become isotopic by one stabilization. On the other hand, we construct infinitely many examples such that cl (S3 - N(K # K′ ∪ t1 ∪ t′1 ∪ t′2)) is not a genus four handlebody and give candidates of two splittings which cannot be made isotopic by a single stabilization.

INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

Let M be S3, S1 × S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.

TUNNEL NUMBER ONE KNOTS HAVE PALINDROME PRESENTATIONS

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set [Formula: see text].