The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two

Abstract

We exhibit an infinite family of knots in the Poincaré homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincaré homology sphere with a lens space surgery has either no symmetries or just a single strong involution. − 3n 2 +n+1 −3n+2 . Through double branched coverings and the Montesinos Trick, the arc κ n lifts to a knot K n in the Poincaré homology sphere on which an integral surgery yields the lens space L(3n 2 + n + 1, −3n + 2).As presented, the arc κ n lies in a bridge sphere that presents P (−2, 3, 5) as a 3-bridge link. Thus the knot K n lies in a genus 2 Heegaard surface for the Poincaré homology sphere P. Nevertheless, as we shall see, generically K n does not have tunnel number one.Theorem 1. For each integer n, the knot K n in the Poincaré homology sphere P has an integral surgery to the lens space L(3n 2 + n + 1, −3n + 2). If |n| ≥ 4, then K n has tunnel number 2.Previously, the only known knots in P admitting a surgery to a lens space are surgery duals to Tange knots [Tan09a] and certain Hedden knots [Hed11, Ras07, Bak14b]. (See Section 5 for a discussion of the Hedden knots.) These knots in P are all doubly primitive, they may be presented as curves in a genus 2 Heegaard splitting that represent a generator in π 1 of each handlebody bounded by the Heegaard surface. We see this as Tange knots and Hedden knots all admit descriptions by doubly pointed genus 1 Heegaard diagrams.(Tange knots are all simple knots in lens spaces and Hedden knots are "almost simple".) Equivalently, they are all 1-bridge knots in their lens spaces. Any integral surgery on a 1-bridge knot in a lens space naturally produces a 3-manifold with a genus 2 Heegaard surface in which the dual knot sits as a doubly primitive knot.The Berge Conjecture posits that any knot in S 3 with a lens space surgery is a doubly primitive knot [Ber], cf. [Gor91, Question 5.5]. It is also proposed that any knot in S 1 × S 2 with a lens space surgery is a doubly primitive knot [BBL13, Conjecture 1.1], [Gre13, Conjecture 1.9]. By analogy and due to a sense of simplicity 1 , one may suspect that any knot in P with a lens space surgery should also be doubly primitive. However this is not the case. Since our knots K n have tunnel number 2 for |n| ≥ 4, they cannot be doubly primitive.Corollary 2. There are knots in the Poincaré homology sphere with lens space surgeries that are not doubly primitive. That is, the analogy to the Berge Conjecture fails for the Poincaré homology sphere.Corollary 3. Conjecture 1.10 of [Gre13] is false. For example, the lens space L(5, 1) contains a knot K * 1 with a surgery to P but does not contain a Tange knot or a Hedden knot with a P surgery.Proof. For n = 1, surgery on K 1 produces L(5, −1) as shown in Figure 1. A quick check of [Tan09a, Table 2] shows that L(5, −1) contains no Tange knots. The Hedden knots with surgery to ±P are homologous to Berge ...

Distance one lens space fillings and band surgery on the trefoil knot

“…We claim that there is no Spin c structure on L(−3k, 1) compatible with N t,1 = k 2 or k−2 2 in (7). Suppose for the contrary such a Spin c structure exists corresponding to j ∈ Z/3k.…”

“…Consider the statement of Proposition 2.9 in the case that t is self-conjugate on L(3, 1) and i = 0 on L(k, 1). We would like to determine which Spin c structure on L(−3k, 1) is induced by (7). As in the previous cases, when k is odd, t 0 is the unique self-conjugate Spin c structure on L(−3k, 1), which corresponds to 0.…”

“…Dehn fillings that are distance one from the fiber slope of a cable space are especially prominent in surgeries yielding prism manifolds [9]. Fillings distance one from the meridional slope were also exploited in [7] to construct cyclic surgeries on knots in the Poincaré homology sphere.…”