The length of unknotting tunnels

Unknotting tunnels and Seifert surfaces

Let $K$ be a knot with an unknotting tunnel $γ$ and suppose that $K$ is not a 2-bridge knot. There is an invariant $ρ= p/q \in \mathbb{Q}/2 \mathbb{Z}$, $p$ odd, defined for the pair $(K, γ)$. The invariant $ρ$ has interesting geometric properties: It is often straightforward to calculate; e. g. for $K$ a torus knot and $γ$ an annulus-spanning arc, $ρ(K, γ) = 1$. Although $ρ$ is defined abstractly, it is naturally revealed when $K \cup γ$ is put in thin position. If $ρ\neq 1$ then there is a minimal genus Seifert surface $F$ for $K$ such that the tunnel $γ$ can be slid and isotoped to lie on $…

On the intersection of unknotting tunnels and the decomposing annulus in connected sums

Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot $K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is a minimal collection of disks we study the components of $V_1-N(A)$ and show that at most one component is a 3-ball meeting $A$ in two disks. This is a crucial step in proving the conjecture that a necessary and sufficient condition for the tunnel number of a connected sum to be less than or equal to the sum of the tunnel numbers is that one of the knots has a H…

Dehn filling and the geometry of unknotting tunnels

Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of tau relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma, and Weeks. We also construct an explicit sequence of one-tunnel knot…

Tunnel One, Fibered Links

Let K be a tunnel number one, fibered link in S^3, with fiber F, and unknotting tunnel $τ$. We show that $τ$ can be isotoped to lie in F.

Twisted torus knots T(p,q,3,s) are tunnel number one

We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, who…

Geodesic systems of tunnels in hyperbolic 3-manifolds

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a system of at least two tunnels, such that all but one of the tunnels come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number n manifolds. In order to show this result, we prove there is …

Classification of alternating knots with tunnel number one

This paper gives a complete classification of all alternating knots with tunnel number one, and all their unknotting tunnels. We prove that the only such knots are two-bridge knots and certain Montesinos knots.

Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots

A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However the converse is not true in general. Then we study such general case in this paper. One of the conclusions is that the unknotting tunnel may be levelled with the torus \partial V_1=\partial V_2.

On Heegaard splittings of knot exteriors with tunnel number degenerations

Let $K_1, K_2$ be two knots with $t(K_1)+t(K_2)>2$ and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a complete classification of those four genus three Heegaard splittings and show unknotting tunnel systems of knots $K_1 # K_2$ corresponding to those Heegaard splittings.

Distinguished waves and slopes in genus two

If R is a nonseparating simple closed curve on the boundary of a genus two handlebody H and H[R] has incompressible boundary, then there exists a unique arc omega in bdry(H), meeting R only in its endpoints, such that, omega is isotopic in bdry(H), keeping its endpoints on R, to a nontrivial wave based at R in each Heegaard diagram of R on bdry(H) which has no cut-vertex. Then surgery on R along its "distinguished-wave" omega yields a pair of simple closed curves, say m_1 and m_2, in bdry(H), each representing an "omega-determined-slope" m on bdry(H[R]), that depends only on R and H. A few con…

Band taut sutured manifolds

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. We use this result to show that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a ce…

Waist size for cusps in hyperbolic 3-manifolds II

The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In this paper, it is proved that the next two smallest waist sizes are realized uniquely for the cusps in the $5_2$ knot complement and the manifold obtained by (2,1)-surgery on the Whitehead link. One application is an improvement on the universal upper bound for the length of an u…

Knot homotopy in subspaces of the 3-sphere

We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighborhoods of embedded graphs, where a knot in a 3-subspace of $S^3$ is said to be transient if it can be moved by a homotopy within the subspace to the trivial knot in $S^3$. To show this, we discuss relation between certain group-theoretic and homotopic properties of knots in a comp…

双曲流形中的伸缩节

Expansion joints in hyperbolic manifolds

研究双曲流形通过特定度量变形，将奇异2 - 柄胶合到秩1尖点，通过操纵基本多面体改变锥角实现插值，应用于算术流形锥变形，证明可钻出2 - 桥链上解结隧道，且结构自然出现在完全增强链中。