C. McA. Gordon

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

Reducing heegaard splittings

Abstract If a Heegaard splitting of a nonsufficiently large 3-manifold has the property that there exist essential disks, one in each of the two Heegaard handlebodies, whose boundaries are disjoint, then the splitting is reducible.

Incompressible planar surfaces in 3-manifolds

Abstract Let M be an orientable 3-manifold and T a torus component of ∂ M . We show that the boundary-slopes of incompressible, boundary-incompressible planar surfaces ( P ,∂ P )⊂( M , T ) are pairwise within distance 4; in particular, there are at most six such boundary-slopes. A corollary is that, for any knot K in S 3 , at most six Dehn surgeries on K can yield a reducible 3-manifold.

On primitive sets of loops in the boundary of a handlebody

Abstract A set C of disjoint simple loops in the boundary of a handlebody X is primitive (that is, geometrically dual to the boundaries of disjoint disks in X ) if and only if adding 2-handles to X along any subset of C yields a handlebody. Also, a set C of n + 1 disjoint simple loops in the boundary of a handlebody X of genus n is standard, in the sense that the loops cobound a planar surface P in ∂ X such that ( X , P ) ≅ ( P × I , P ×{0}), if and only if adding 2-handles to X along any proper subset of C yields a handlebody.

Knots, homology spheres, and contractible 4-manifolds

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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.

A loop theorem for duality spaces and fibred ribbon knots

In this paper we prove a version of the loop theorem for surfaces in the boundary of a 3-dimensional duality space, i.e. a space which resembles a 3manifold only in that it satisfies the appropriate form of Poincar6-Lefschetz duality over some field of untwisted coefficients. Our motivation comes from the fact that such spaces occur as the infinite cyclic coverings of certain 4manifolds which arise in the study of knot concordance, and as the main application of our theorem we show that if a fibred knot in the 3-sphere is a ribbon knot, then its monodromy extends over a handlebody. We approach the loop theorem via the study of planar coverings of a surface, as in the original paper of Papakyriakopoulos [11] and the subsequent work of Maskit [9]. w contains a simple geometric treatment of these matters. The main result of w is that a duality space actually satisfies duality with (twisted) coefficient module the quotient of the fundamental group ring by any power of the augmentation ideal. In w 4, the results of w167 2 and 3, together with an algebraic lemma on the intersection of the powers of the augmentation ideal of a group ring, are used to prove the loop theorem for 3-dimensional duality spaces. (For the readers convenience a proof of the algebraic temma is included as an appendix.) w contains the application to fibred ribbon knots mentioned above. In w the result of w is used to obtain a limited amount of information on some questions about knots in the boundaries of contractible 4-manifolds. In w we apply our methods to another aspect of knot concordance, and show that for any concordance with a rationally anisotropic fibred knot (see [7]) at one end, the inclusion of the complement of the knot into the complement of the concordance induces an injection of fundamental groups. For torus knots, this question was raised by Scharlemann [14].

Chapter VII Incompressible Surfaces in Branched Coverings

Publisher Summary This chapter discusses the incompressible surfaces in branched coverings. All the manifolds—including surfaces—are assumed to be orientable. The three-manifolds are generally assumed to be connected. A link in a closed three-manifold is a closed one-dimensional submanifold. A link is trivial if it consists of one component and bounds a disk. A link is prime if it is not a connected sum of two nontrivial links. An extension of Hakens finiteness theorem applied to annuli implies that every link is a finite connected sum of prime links. A three-manifold is hyperbolic if it has a complete hyperbolic structure with finite volume. The chapter applies an extension of Hakens finiteness theorem applied to annuli to imply that every link is a finite connected sum of prime links. The chapter describes equivariant loop theorem for involutions.

Toroidal Dehn surgeries on knots in lens spaces

Let M be a compact, orientable 3-manifold with ∂ M a torus. If r is a slope on ∂ M (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold M ( r ) by gluing a solid torus V r to M along their boundaries in such a way that r bounds a disc in V r . We say that M ( r ) is obtained from M by r -Dehn filling. Assume now that M contains no essential sphere, disc, torus or annulus. Then, by Thurstons Geometrization Theorem for Haken manifolds [ T1 , T2 ], M is hyperbolic, in the sense that int M has a complete hyperbolic structure of finite volume. Furthermore, M ( r ) is hyperbolic for all but finitely many r [ T1 , T2 ] and the precise nature of the set of exceptional slopes E ( M )={ r : M ( r ) is not hyperbolic} has been the subject of a considerable amount of investigation. The maximal observed value of e ( M )=[mid ] E ( M )[mid ] (the cardinality of E ( M )) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [ T1 ]. Let Δ( r 1 , r 2 ) denote as usual the minimal geometric intersection number of two slopes r 1 and r 2 . If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ( r 1 , r 2 ): r 1 , r 2 ∈[Sscr ]} gives one for [mid ][Sscr ][mid ]. For example, one can check (using [ GLi , lemma 2·1]) that for 1[les ]Δ([Sscr ])[les ]10, the maximum values of [mid ][Sscr ][mid ] are as given in Table 1. In particular, any upper bound for Δ( M )=Δ( E ( M )) gives a corresponding bound for e ( M ). (The maximal observed value of Δ( M ) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [ T1 , HW ].) If M ( r ) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base S 2 and at most three singular fibres), or a counterexample to the Geometrization Conjecture [ T1 , T2 ]. A survey of the presently known upper bounds on the distances Δ( r 1 , r 2 ) between various classes of exceptional slopes r 1 and r 2 , and the maximal values realized by known examples, is given in [ Go2 ]. (See also [ Wu2 ] for a discussion of the additional cases that arise when M has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.

A short proof of a theorem of Plans on the homology of the branched cyclic coverings of a knot

Let KQS be a (tame) knot, with complement C = S—i£, and let C be the infinite cyclic covering of K, i.e. the covering of C corresponding to the commutator subgroup of xi(C). The group of covering translations of C is iTi(C), which is infinite cyclic by Alexander duality; this gives an action of Zon i?i(C), and so H\{C) becomes a Amodule, where A is the integral group ring of Z. We identify A with the ring of polynomials in a single variable /, (positive and negative powers of t being allowed), with integral coefficients. (See [4].) The &-fold branched cyclic covering of K, Mk (k^l) is defined by taking the covering of C corresponding to the kernel of the composition: