Dehn Surgery Research Articles

Dehn surgery on knots in solid tori creating essential annuli

Let M M be a 3 3 -manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when M M contains a separating essential annulus. It is shown that M M does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the 3 3 -sphere which are not periodic knots of period 2 2 satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the 2 2 -sphere with exactly three exceptional fibres.

Dehn surgery on knots in 3-manifolds

It has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in S3 [8]. However, a classification of such 3-manifolds in terms of this surgery construction has remained elusive. This is due primarily to the lack of uniqueness of the surgery description. In [5], Kirby gave us a calculus of surgery diagrams. However, the lack of a 'canonical' surgical method of getting from one 3-manifold to another has hampered further progress. In this paper, we show that if one restricts attention to the case where a surgery curve is homotopically trivial in a 3-manifold, then one has the following uniqueness theorem.

Invariants of Three-Manifolds, Unitary Representations of the Mapping Class Group, and Numerical Calculations

Based on previous results of the two first authors, it is shown that the combinatorial construction of invariants of compact, closed three-manifolds by Turaev and Viro as state sums in terms of quantum 6j-symbols for SLq(2; C) at roots of unity leads to the unitary representation of the mapping class group found by Moore and Seiberg. Via a Heegaard decomposition this invariant may therefore be written as the absolute square of a certain matrix element of a suitable group element in this representation. For an arbitrary Dehn surgery on a figure-eight knot we provide an explicit form for this matrix element involving just one 6j-symbol. This expression is analyzed numerically and compared with the conjectured large k = r – 2 asymptotics of the Chern–Simons–Witten state sum [Witten 1989], whose absolute square is the Turaev-Viro state sum. In particular we find numerical agreement concerning the values of the Chern–Simons invariants for the flat SU(2)-connections as predicted by the asymptotic expansion of the statesum with analytical results found by Kirk and Klassen [1990].

Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebodies

Given a knot K in a 3-manifold M, we use N(K) to denote a regular neighbourhood of K. Suppose γ is a slope (i.e. an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H – Int N(K) so that γ bounds a meridional disc. We say that M is ∂-reducible if ∂M is compressible in M, and we call γ a ∂-reducing slope of K if (H, K; γ) is ∂-reducible. Since incompressible surfaces play an important rôle in 3-manifold theory, it is important to know what slopes of a given knot are ∂-reducing. In the generic case there are at most three ∂-reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂-reducing slopes only if it is a 1-bridge braid. Moreover, all such knots and their corresponding ∂-reducing slopes are classified in [1]. For 1-bridge braids with small bridge width, a geometric method of detecting ∂-reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e. if K is a knot in a handlebody with H – K ∂-irreducible, then K has ∂-reducing slopes only if K is a 1-bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems.

Dehn surgeries and [formula omitted]2-reducible 3-manifolds

This note concerns 3-manifolds M obtained by Dehn surgery on a knot in S 3, in particular those which contain embedded projective planes. Generically, in the knot-space S 3 − N ( k), the intersection of a projective plane P 2 in M, and any 2-sphere S 2 in S 3 pierced by k, is a 1-complex which can be viewed as a graph in either the projective plane or the 2-sphere. C.McA. Gordon and J. Luecke have used similar graphs arising as the intersection of two 2-spheres to prove that a knot in S 3 is determined by its complement. With a view to adapting their techniques to show that P 3 cannot be obtained by Dehn surgery on a knot in S 3, we prove here that a “minimal” P 2 in M (i.e., a P 2 which minimizes the odd number of intersections with the core of the solid torus added during surgery) gives rise to a graph (in S 2), which contains no “generalized” Scharlemann cycles. This obstruction is considered as a step towards showing that P 3 cannot be obtained by Dehn surgery along a knot in S 3.

Dehn surgery and essential annuli

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

Reducible manifolds and Dehn surgery

Reducible manifolds and Dehn surgery

A note on strong geometric isolation in 3-orbitfolds

Neuman and Reid describe a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' “SnapPea” program, they conjecture that the cusps are strongly geometrically isolated, a fact which we establish here. We also give a parameterisation of the Dehn Surgery Space of this orbifold which has amusing properties.

Finite Dehn surgery on knots

Let K be a knot with a closed tubular neighbourhood N(K) in a connected orientable closed 3-manifold W , such that the exterior of K, M = W − intN(K), is irreducible. We consider the problem of which Dehn surgeries on K, or equivalently, which Dehn fillings on M , can produce 3-manifolds with finite fundamental group. For convenience, a surgery is called a G-surgery if the resultant 3-manifold has fundamental group G. If G is cyclic or finite, the surgery is also called a cyclic surgery or a finite surgery. Similar terminology will be used for Dehn fillings. The manifold obtained by the Dehn filling on M along ∂M with slope r is denoted by M(r). Let ∆(r1, r2) denote the minimal geometric intersection number (the distance) between two slopes r1 and r2 on ∂M . According to [T1], M belongs to one of the following three mutually exclusive categories: (I) M is a Seifert fibred space admitting no essential torus. (II) M is a hyperbolic manifold (i.e. intM admits a complete hyperbolic metric of finite volume). (III) M contains an essential torus. It is a remarkable result, the so-called cyclic surgery theorem [CGLS], that if M is not Seifert fibred, then all cyclic surgery slopes of K have mutual distance no larger than 1, and consequently, there are at most 3 cyclic surgeries on K. In this paper, we consider finite Dehn surgery on K and we prove, for instance, that if M is not a manifold of type (I) and is not a union along a torus of the twisted I-bundle over the Klein bottle and a cabled space, then there are at most 6 finite and cyclic surgeries on K of maximal mutual distance 5. Henceforth, we shall use finite/cyclic to mean either finite or (infinite) cyclic. It turns out to be convenient to consider the three cases mentioned above separately. In case (I) it is well-known that one can completely classify the finite/cyclic surgeries on K. Considering the torus knots for instance, one sees that there exist infinitely many knots whose exteriors are of type (I), each of which admit an infinity of distinct finite (cyclic or non-cyclic) surgery slopes. Our contributions deal with the cases (II) and (III). For the former we obtain

On the eta-invariant of some hyperbolic 3-manifolds

We derive an explicit formula for the η-invariant of hyperbolic 3-manifolds obtained by Dehn surgeries on some one-cusped hyperbolic 3-manifolds (e.g., hyperbolic knot complements in S 3).

Essential laminations and Dehn surgery on 2-bridge knots

In this paper we show that any nontrivial Dehn surgery on a nontorus 2-bridge knot produces a manifold which is covered by R 3. In particular, this manifold is irreducible and has infinite fundamental group. (As a consequence, it is also clear that 2-bridge knots satisfy property P, although this was shown previously by Takahashi (1981) using more algebraic techniques.) The result is a consequence of showing that such a manifold is laminar, that is it contains an essential lamination. We accomplish this by constructing in the exterior of each nontorus 2-bridge knot an essential lamination which remains essential in all manifolds produced by nontrivial Dehn filling. We call an essential lamination with this property persistent. The examples of essential laminations produced in this way are of interest since they tend support the conjecture that “most” manifolds with infinite fundamental group contain essential laminations.

Arf invariants of strongly invertible knots obtained from unknotting number one knots

Two techniques, branched covering space and Dehn surgery, are known as popular methods by which we obtain new 3-manifolds from a link in S. In 1977, Montesinos [8] showed the following relationship between the 2-fold cyclic branched covering of S branched over a link and the closed, orientable 3-manifold which is obtained by doing surgery on a link in S. A link in S is called strongly invertible if there is an orientation preserving involution of S which induces in each component of L an involution with two fixed points.

Dehn surgeries on knots creating essential tori, I

Dehn surgeries on knots creating essential tori, I

A Dehn Surgery Formula for Walker Invariant on a Link

A Dehn Surgery Formula for Walker Invariant on a Link

Homology lens spaces and Dehn surgery on homology spheres

A homology lens space is a closed 3-manifold with ℤ-homology groups isomorphic to those of a lens space. A useful theorem found in [Fu] states that a homology lens space $M^3$ may be obtained by an (n/1)-Dehn surgery on a homology 3-sphere if and only if

On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows

On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows

?-reducing Dehn surgeries and 1-bridge knots

?-reducing Dehn surgeries and 1-bridge knots

Topological field theory and surgery on three-manifolds

The solution of the SU(2) quantum Chern-Simons field theory defined on a closed, connected and orientable three-manifold is presented. The vacuum expectation values of Wilson line operators, associated with framed links in a generic manifold, are computed in terms of the expectation values of the three-sphere. The method consists of using an operator realization of Dehn surgery. The rules, corresponding to the surgery instructions in the three-sphere, are derived and the three-manifold invariant defined by the Chern-Simons theory is constructed. Several examples are considered and explicit results are reported.

Alternating knots and non-Haken 3-manifolds

It is proved in this paper that there is an infinity of knot types in S 3, having essential closed embedded surfaces in their exteriors but yielding non-Hakem manifolds by Dehn surgery for infinitely many surgery coefficients.

Dehn fillings of Klein bottle bundles

An important problem in the topology of 3-manifolds is to classify manifolds obtained by Dehn surgeries on a knot in a closed 3-manifold, or equivalently, Dehn fillings of a 3-manifold M with boundary a torus.