John Luecke

Dehn Surgery on Knots

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that

Toroidal and boundary-reducing Dehn fillings

Abstract Let M be a simple 3-manifold with a toral boundary component ∂ 0 M . If Dehn filling M along ∂ 0 M one way produces a toroidal manifold, and Dehn filling M along ∂ 0 M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on ∂ 0 M of the two filling slopes is at most two. In the special case that the boundary-reducing filling is actually a solid torus and the intersection number between the filling slopes is two, more is said to describe the toroidal filling.

Finite covers of 3-manifolds containing essential tori

It is shown in this paper that if a Haken 3-manifold contains an incompressible torus that is not boundary-parallel then either it has a finite cover that is a torus-bundle over the circle or it has finite covers with arbitrarily large first Betti number. In [He 4], Hempel conjectures that every Haken 3-manifold has a finite cover whose fundamental group has a nontrivial representation to the integers (i.e. the group is indicable). The conjecture is proved in this paper in the case that the Haken manifold contains an incompressible torus. In particular, it is shown (Theorem 1.1) that if Haken manifold contains an incompressible torus that is not boundaryparallel then either it has a finite cover that is a torus-bundle over the circle or it has finite covers with arbitrarily large first Betti number. Hempels conjecture arises in the context of the following question: (1) does every irreducible 3-manifold with infinite fundamental group have a finite cover which is Haken? An affirmative answer to this would be a big step toward the classification of 3-manifolds with infinite fundamental group. How would one detect such a cover? A common way of showing that an irreducible 3-manifold is Haken is to show that its fundamental group has a nontrivial representation to the integers, then use this fact to find a map from the 3-manifold to the circle for which the pre-image of some regular value is an incompressible surface. Thus question (1) leads to: (2) does every irreducible 3-manifold have a finite cover whose fundamental group has a nontrivial representation to the integers? An affirmative answer to (2) would certainly imply one for (1); however, the converse is not known. In fact, the conjecture of Hempel stated in the first paragraph says exactly that (1) implies (2). For example, let M be the union of two knot complements identified along their boundaries in such a way that the meridian of each knot complement is identified with the longitude of the other. Then M is Haken, but 7Ti (M) has no nontrivial representation to the integers (Hy(M) = 0). However, it is shown in [He 4] that in many cases M will have a finite cover whose fundamental group does have such a representation. Note that there M contains an incompressible torus that is not boundary parallel, hence Theorem 1.1 applies. In fact, the argument used to prove Theorem 1.1 is a generalization of that used in [He 4] for this example. This paper is taken from part of the authors Ph.D. dissertation and as such owes much to Cameron Gordon. Received by the editors February 20, 1986 and, in revised form, September 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M05, 57M10; Secondary 20E07. This work was supported in part by NSF grant DMS-8504683. ©1988 American Mathematical Society 0002-9947/88

Knots with unknotting number 1 and essential Conway spheres

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Dehn Surgery on Knots in the 3-Sphere

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Tangle analysis of difference topology experiments: Applications to a Mu protein-DNA complex

For a knot K in S 3 , let T.K/ be the characteristic toric sub-orbifold of the orbifold .S 3 ;K/ as defined by Bonahon‐Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T.K/, unless either K is an EM‐knot (of Eudave-Munoz) or .S 3 ;K/ contains an EM‐tangle after cutting along T.K/. As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsvath‐Szabo, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.

Bridge number and integral Dehn surgery

The Dehn surgery construction is a way of obtaining a closed 3-manifold from a knot in the 3-sphere. The construction depends on two parameters, the knot and the surgery slope, and this article discusses theorems and conjectures describing the way the topology and geometry of the 3-manifold constructed depend on these knots in arbitrary 3-manifolds and the Dehn surgery construction there. Most of the theorems discussed here apply in that context as well. [Go1] is an excellent survey at this level and I recommend it as a companion to this article. My intent here is to update some of the issues in [Go1] as well as to draw attention to some tantalizing aspects of specializing to knots in the 3-sphere.

Bridge number, Heegaard genus and non-integral Dehn surgery

We develop topological methods for analyzing difference topology experiments involving 3‐string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number. 57M25, 92C40

Links with unlinking number one are prime

R and give a lower bound on the bridge number of K n with respect to Heegaard splittings of M; as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of K n tends to infinity with n. In application, we look at a family of knotsfK n g found by Teragaito that live in a small Seifert fiber space M and where each K n admits a Dehn surgery giving S 3 . We show that the bridge number of K n with respect to any genus-2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving S 3 .

Reducible manifolds and Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.