Nontrivial Surgery Research Articles

A Quasi-Wireless Intraoperatory Neurophysiological Monitoring System

Intraoperative Neurophysiological Monitoring is a set of monitoring techniques that reads electrical activity generated by the nervous system structures during surgeries. In non-trivial surgeries, neurophysiologists require a significant number of electrical signals to be picked up to check the effects of the surgeon’s actions in real time or to confirm that the correct nerves are selected. As a result, cabling the patient in the operating room can become cumbersome. The proposed WIONM module solves part of the problem by converting a good part of those cables into a wireless connection that is substantially transparent to the human operator and the existing medical instrumentation.

Anosov flows on -manifolds: the surgeries and the foliations

AbstractEvery Anosov flow on a 3-manifold is associated to a bifoliated plane (a plane endowed with two transverse foliations $F^s$ and $F^u$ ) which reflects the normal structure of the flow endowed with the center-stable and center-unstable foliations. A flow is $\mathbb{R}$ -covered if $F^s$ (or equivalently $F^u$ ) is trivial. On the other hand, from any Anosov flow one can build infinitely many others by Dehn–Goodman–Fried surgeries. This paper investigates how these surgeries modify the bifoliated plane. We first observe that surgeries along orbits corresponding to disjoint simple closed geodesics do not affect the bifoliated plane of the geodesic flow of a hyperbolic surface (Theorem 1). Analogously, for any non- $\mathbb{R}$ -covered Anosov flow, surgeries along pivot periodic orbits do not affect the branching structure of its bifoliated plane (Theorem 2). Next, we consider the set $\mathcal{S}urg(A)$ of Anosov flows obtained by Dehn–Goodman–Fried surgeries from the suspension flow $X_A$ of any hyperbolic matrix $A \in SL(2,\mathbb{Z})$ . Fenley proved that performing only positive (or negative) surgeries on $X_A$ leads to $\mathbb{R}$ -covered Anosov flows. We study here Anosov flows obtained by a combination of positive and negative surgeries on $X_A$ . Among other results, we build non- $\mathbb{R}$ -covered Anosov flows on hyperbolic manifolds. Furthermore, we show that given any flow $X\in \mathcal{S}urg(A)$ there exists $\epsilon>0$ such that every flow obtained from $X$ by a non-trivial surgery along any $\epsilon$ -dense periodic orbit $\gamma$ is $\mathbb{R}$ -covered (Theorem 4). Analogously, for any flow $X \in \mathcal{S}urg(A)$ there exist periodic orbits $\gamma_+,\gamma_-$ such that every flow obtained from $X$ by surgeries with distinct signs on $\gamma_+$ and $\gamma_-$ is non- $\mathbb{R}$ -covered (Theorem 5).

Knot complement problem for L-space ℤHS3

In this paper, we look at the knot complement problem for L-space [Formula: see text]-homology spheres. We show that an L-space [Formula: see text]-homology sphere [Formula: see text] cannot be obtained as a non-trivial surgery along a knot [Formula: see text]. As a consequence, we prove that knots in an L-space [Formula: see text]-homology sphere are determined by their complements.

Surgery on a knot in (surface×I)

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.

Links with surgery yielding the 3-sphere

For any n≥2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. In particular, our links have tunnel number n-1. We can also give infinitely many 2-component hyperbolic links with such properties.

Relations of smooth Kervaire classes over the mod 2 Steenrod algebra

In the surgery theory of smooth smooth manifolds, it is often difficult to determine the existence or the non-existence of a smooth normal map with nontrivial surgery obstructions. We present and prove a simple relation over the mod 2 Steenrod algebra between two smooth Kervaire classes in different dimensions. This formula enables us to compute the Kervaire surgery invariants for various manifolds.

Incompressible surfaces in link complements

We generalize a theorem of Finkelstein and Moriah and show that if a link L has a 2n-plat projection satisfying certain conditions, then its cornplement contains some closed essential surfaces. In most cases these surfaces remain essential after any totally nontrivial surgery on L. A link L in S3 has a 2n-plat projection for some n, as shown in Figure 1, where a box on the i-th row and j-th column consists of 2 vertical strings with an aij left-hand half twist; in other words, it is a rational tangle of slope 1/aij. See for example [BZ]. Let n be the number of boxes in the even rows, so there are n 1 boxes in the odd rows. Let m be the number of rows in the diagram. It was shown by Finkelstein and Moriah [FMi], [FM2] that if n > 3, m > 5, and if laijI > 3 for all i, j, then the link exterior E(L) = S3 IntN(L) contains some essential planar surfaces, which can be tubed on one side to obtain closed incompressible surfaces in E(K). In this note we will prove a stronger version of this theorem, showing that E(L) contains some essential surfaces if n > 3, the boxes at the two ends of the odd rows have IaijI > 3, and aij 0 for the boxes which are not on the ends of the rows. We allow aij = 0 for boxes at the ends of the even rows, and there is no restriction on m, the number of rows in the diagram. The argument here provides a much simpler proof to the above theorem of Finkelstein and Moriah. In [FM2] that theorem was applied to show that if L is a knot, then all surgeries on L contain essential surfaces. Corollary 2 below generalizes this to the case when L has multiple components, with a mild restriction that each component of L intersects some spheres. We first give some definitions. Let a = a(al,... , am) be an arc running monotonically from the top to the bottom of the 2n-plat, such that ae. is disjoint from the boxes, and on the i-th row there are ai boxes on the left of oa. See Figure 1 for the arc ai(1, 1, 1, 2, 2). The arc ae is an allowable path if (i) each row has at least one box on each side of oa, and (ii) ae intersects L at m + 1 points (so ae intersects L once when passing from one row to another). Note that the leftmost allowable path is a(1,... , 1), which has on its left one box from each row. Given an allowable path a = a(ai,... , am), we can connect the two ends of ae by an arc 13 disjoint from the projection of L to form a circle, then cap it off by two disks, one on each side of the projection plane, to get a sphere S = S(ai,... , am), Received by the editors February 22, 2000 and, in revised form, March 27, 2000. 1991 Mathematics Subject Classification. Primary 57N10, 57M25.

ESSENTIAL ARCBODY AND TANGLE DECOMPOSITIONS OF KNOTS AND LINKS

We prove that if K⊂ S3 is either: (I) a link with an essential n-string arcbody decomposition, where at least one arcspace has incompressible boundary, or a knot with an essential n-string tangle decomposition, where (II) each tangle has no parallel strings, or (III) one tangle space is not a handlebody and K is not cabled, then any nontrivial surgery on every component of K produces irreducible manifolds in all cases (with some exceptional surgeries in case (I)) and, in particular, Haken manifolds in cases (I) and (III). Moreover, if K is hyperbolic in (III) and at least one tangle space has incompressible boundary, then all nontrivial surgeries on K are also hyperbolic; this last result is also established for type (I) decompositions under some constraints.

Tubed incompressible surfaces in knot and link complements

We prove that the complements of all knots and links in S 3 which have a 2n-plat projection with absolute value of all twist coefficients bigger than 2 contain closed embedded incompressible non-boundary parallel surfaces. These surfaces are obtained from essential planar meridional surfaces by tubing to one side along the knot or link. In the case of a knot it follows that these surfaces stay incompressible in all manifolds obtained by non-trivial surgery on the knot.

Dehn surgery on arborescent links

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link L L is sufficiently complicated, in the sense that it is composed of at least 4 4 rational tangles T ( p i / q i ) T(p_{i}/q_{i}) with all q i > 2 q_{i} > 2 , and none of its length 2 tangles are of the form T ( 1 / 2 q 1 , 1 / 2 q 2 ) T(1/2q_{1}, 1/2q_{2}) , then all complete surgeries on L L produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let T ( r / 2 s , p / 2 q ) = ( B , t 1 ∪ t 2 ∪ K ) T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K) be a tangle with K K a closed circle, and let M = B − Int ⁡ N ( t 1 ∪ t 2 ) M = B - \operatorname {Int} N(t_{1}\cup t_{2}) . We will show that if s > 1 s>1 and p ≢ ± 1 p \not \equiv \pm 1 mod 2 q 2q , then ∂ M \partial M remains incompressible after all nontrivial surgeries on K K . Two bridge links are a subclass of arborescent links. For such a link L ( p / q ) L(p/q) , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless p / q p/q has a partial fraction decomposition of the form 1 / ( r − 1 / s ) 1/(r-1/s) , in which case it does admit non-laminar surgeries.

Symmetry of knots and cyclic surgery

If a nontorus knot K K admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on K K . No surgery on a symmetric knot can produce a fake lens space or a 3 3 -manifold M M with | π 1 ( M ) | = 2 |{\pi _1}(M)|= 2 . This generalizes the result of Culler-Gordon-Luecke-Shalen-Bleiler-Scharlemann and supports the conjecture that no nontrivial surgery on a nontrivial knot yields a 3 3 -manifold M M with | π 1 ( M ) | > 5 |{\pi _1}(M)| > 5 .

Cyclic surgery on satellite knots

In [9] L. Moser classified all manifolds obtained by Dehn surgery on torus knots. In particular she proved the following (see also [8, Chapter IV]).Theorem 1 [9]. Nontrivial surgery with slope m/n on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iff m = npq ± 1 and the manifold obtained is the lens space L(m, nq2).

Pseudo-Anosov maps and surgery on fibred 2-bridge knots

We study fibred knots by computing invariant laminations of the monodromy of the fibration and then computing the degeneracy type of the knot. We show that nontrivial surgery on a nontorus fibred 2-bridge knot yields a manifold whose universal cover is R 3.

Producing reducible 3-manifolds by surgery on a knot

IT HAS long been conjectured that surgery on a knot in S3 yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [7.8, 9, 10, 111). One may regard the Poenaru conjecture (solved in [S]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold A4 produces a reducible 3-manifold M’. But this problem is too complex, since, dually, it asks which knots in which manifolds arise from surgery on reducible 3-manifolds. In this paper we are able to show, approximately, that if M itself either contains a summand not a rational homology sphere or is a-reducible, and M’ is reducible, then k must have been cabled and the surgery is via the slope of the cabling annulus. Thus the result stops short of proving the conjecture for M = S3, but (see below) does suffice to prove the conjecture for satellite knots. The results here are broader than this; for a context recall the main result of [3]: GABAI’S THEOREM. Let k be a knot in M = D2 x S’ with nonzero wrapping number. If_ci’ is a manifold obtained by non-trivial surgery on k then one of the following must hold: (1) M = D2 x S’ = M’ and both k and k’ are 0 or l-bridge braids. (2) M’ = WI # W,, where W2 is a closed 3-manifold and H, ( W,) isjnite and non-tritial. (3) M’ is irreducible and aM ‘ is incompressible. It is this theorem we generalize to many other manifolds M. We also examine case (2) and show that it only arises (i.e. M’ is only reducible) if k is cabled with cabling annulus having the slope of the surgery. (For a detailed look at case (I), see [6] or Cl].) Specifically we have:

Surgery on knots in solid tori

IN THIS paper we investigate the topology of manifolds obtained by nontrivial surgery on a knot k in D2 x S’. In particular this work together with the work of Berge [l] yields a complete solution to the following old question. Exactly when can one obtain a D’ x S’ by nontrivial surgery on a knot k in D2 x S’? We will assume that k is not contained in a 3-cell in D2 x S’, for otherwise the manifold obtained by surgery on k is a connected sum of D2 x S’ and a manifold obtained by surgery on a knot (i.e. k) in S3. The wrapping number wr(k) of k is equal to the minimal geometric intersection number of a knot k’ with a D2 x pt., where k’ is isotopic to k. wr(k)=O is the same as saying that k is contained in a 3-cell. The winding number w(k) is equal to the algebraic intersection number of k with a D2 x pt. Let T’c D2 x S’ be the torus a(1/2D2) x S1 and let W be the solid torus which it bounds. k is a rorus knot or O-bridge braid if k can be isotoped to lie in T’. k is a lbridge braid if k is isotopic to a knot of the form k’ u k” where k’, k” are arcs, k’ c T’ and is transverse to each D2 x pt. and k” c Wn (D2 x pt.) We now state our main result whose proof is given in $2.

Cyclic surgery on knots

We get a necessary condition under which a nonsimple knot (i.e. a satellite knot) admits a nontrivial surgery producing a lens space. Interesting corollaries are: (1) if a lens space can be obtained from a nontrivial surgery on a nonsimple knot, then the order of its fundamental group is not smaller than 23; (2) any nonsimple knot admits at most one nontrivial surgery which produces a lens space.

Surgery on a class of pretzel knots

Any closed 3-manifold M may be thought of as a union of 3-celIs. Any covering space of M is made up of copies of these 3-cells with boundaries locally identified as in M. Covers of M may be built by piecing together these balls. This paper develops a method to piece together the universal cover of manifolds obtained from the 3-sphere by surgery on a class of pretzel knots in such a way that the cover can be shown to be R3. 1. It has been shown [4, 6, 14] that any connected orientable 3-manifold may be constructed by surgery along a finite number of knots in S 3, that is, by removing tubular neighborhoods of one or more smooth knots and sewing them back in differently. In this paper we study the 3-manifolds obtained by surgery on certain pretzel knots [2, 8] by showing that the universal cover of such manifolds is R3. We thus show that these pretzel knots satisfy several hoped for conjectures (property P: nontrivial surgery never yields a simply connected manifold, and property R: surgery never yields a manifold with fundamental group Z). The manifolds obtained by surgery on pretzel knots have been studied algebraically. In fact, it is known that all pretzel knots (except, of course, the unknot, which has several pretzel knot representations) have property R [5, 7]. The property P question has been answered affirmatively for certain classes of pretzel knots [1, 9, 10, 11], but the question remains open in general. We will take a different, geometric approach to the problem. In §2 we introduce the basic concepts we will use to build the covering spaces of the manifolds, as well as prove a lemma which identifies the covering space in certain situations. In §3 we apply the ideas of §2 to the surgery manifolds of certain pretzel knots and construct their universal covering spaces. 2. In this section we develop concepts which allow us, in certain cases, to build and identify covering spaces inductively. The central idea is precover which isolates properties a space needs in order to be part of a covering space (i.e., an incompletely built covering space).

Surgery on double knots and symmetries

W. Whitten conjectured [Pacific J. Math. 97 (1981), no. 1, 209–216] that no 3-manifold obtained by a nontrivial surgery on a double of a noninvertible knot is a 2-fold branched covering of S3. The authors give counterexamples to this conjecture and determine the exact range of validity of the conjecture. More generally, they consider closed, orientable 3-manifolds obtained by nontrivial Dehn surgery on a double of a non-strongly invertible knot and study the symmetries of such manifolds, i.e. the homeomorphisms of finite order on these manifolds. They show that, except for a finite number of surgeries, these manifolds admit no (nontrivial) symmetry.

Coverings of dehn fillings of surface bundles

This paper concern the study of 3-manifolds which are obtained by Dehn filling on a surface bundle over S 1 (every closed 3-manifold is so representable) and directed toward the question: which 3-manifolds, M, have fundamental group, π 1 ( M), virtually Z -representable (have a finite sheeted covering space M̃ → M with β 1( M ̃ ) = rank H 1( M ̃ )>0) ? For Dehn fillings of a bundle with periodic or reducible monodromy (or of any nonsimple 3-manifold) this is shown to be generally the case with possible exceptions of a specified form. So, for example, nontrivial surgery on a composite knot always produces a 3-manifold with virtually Z -representable fundamental group. We note that any sheeted covering, M̃, of a Dehn filling, M, of a surface bundle, N, is a Dehn filling of a surface bundle, Ñ, covering N. We determine a lower bound for {β 1(M̃): M̃ a filling of Ñ}, and combine these results with some calculations to show that a/ b surgery on the figure eight knot produces manifolds with virtually Z -representable fundamental groups for any a and either b≡±2 a mod 7 or b≡± a mod 13

Left-orderable, non-$L$-space surgeries on knots

Let K be a knot in the 3–sphere S3. An r–surgery on K is leftorderable if the resulting 3–manifold K(r) of the surgery has left-orderable fundamental group, and an r–surgery on K is called an L–space surgery if K(r) is an L–space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be classified into left-orderable surgeries or L– space surgeries. We introduce a way to provide knots with left-orderable, non– L–space surgeries. As an application we present infinitely many hyperbolic knots on each of which every nontrivial surgery is a hyperbolic, left-orderable, non–L–space surgery.