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Every closed orientable 3 3 -manifold is obtained from the 3 3 -sphere S 3 S^3 by Dehn surgery on a link, and thus Dehn surgery is a useful way to construct 3 3 -manifolds. In this survey article we restrict our attention to Dehn surgery on knots in S 3 S^3 and take a quick look at the evolution of study on this field along developments of 3 3 -dimensional topology.

Let K be a nontrivial knot in S 3 with the exterior E ( K ) , and denote π 1 ( E ( K ) ) by G ( K ) . We prove that for any hyperbolic knot K and any nontrivial element g ∈ G ( K ) , there are only finitely many Dehn fillings of E ( K ) which trivialize g. We also demonstrate that there are infinitely many nontrivial elements in G ( K ) which cannot be trivialized by nontrivial Dehn fillings.

Let $K$ be a nontrivial knot in $S^3$ with the exterior $E(K)$, and $\gamma \in G(K) = \pi _1(E(K), *)$ a slope element represented by an essential simple closed curve on $\partial E(K)$ with base point $* \in \partial E(K)$. Since the normal closure $\langle \!\langle \gamma \rangle \!\rangle $ of $\gamma $ in $G(K)$ coincides with that of $\gamma ^{-1}$, and $\gamma $ and $\gamma ^{-1}$ correspond to a slope $r \in \mathbb{Q} \cup \{ \infty \}$, we write $\langle \!\langle r \rangle \!\rangle = \langle \!\langle \gamma \rangle \!\rangle $. The normal closure $\langle \!\langle r \rangle \!\rangle $ describes elements, which are trivialized by $r$-Dehn filling of $E(K)$. In this article, we prove that $\langle \!\langle r_1 \rangle \!\rangle = \langle \!\langle r_2 \rangle \!\rangle $ if and only if $r_1 = r_2$, and for a given finite family of slopes $\mathcal{S} = \{ r_1, \dots , r_n \}$, the intersection $\langle \!\langle r_1 \rangle \!\rangle \cap \cdots \cap \langle \!\langle r_n \rangle \!\rangle $ contains infinitely many elements except when $K$ is a $(p, q)$-torus knot and $pq \in \mathcal{S}$. We also investigate inclusion relation among normal closures of slope elements.

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the (-2, 3, 7)-pretzel knot, have generalized torsion. Since generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups., 14 pages, 14 figures

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m>2) is a generalized torsion element., 14 pages, no figure

Greene has given an infinite family of non-quasi-alternating links L m , n in which L 2 , 3 is homologically thin and its double branched cover X 2 , 3 is an L-space. In this note we show that the double branched cover X m , n of L m , n is an L-space for all n > m ≥ 2 and L m , n is the unique link whose double branched cover is X m , n .

A generalized torsion element is a non-trivial element such that some non-empty finite product of its conjugates is the identity. We construct a generalized torsion element of the fundamental group of a 3-manifold obtained by Dehn surgery along a knot in the 3-sphere., 15 pages, 9 figures. We fixed the problem in figures

The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead doubles of a knot satisfies the Slope Conjecture and the Strong Slope Conjecture if the original knot does. Additionally, we provide a proof that there are Whitehead doubles which are not adequate., This version has been accepted for publication in Quantum Topology. We restructured results and presentation to focus on the maximum degree of colored Jones polynomial for improved exposition and clarity. A cancellation error is also corrected. The changes mainly impact sections 1 and 2

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot., Comment: To appear in Topology Appl

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $3$-manifold $M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $3$-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements., Comment: 10 pages, 2 Figures

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. We further develop this to show that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{ K_n \}$ to contain infinitely many L-space knots, and show (modulo a conjecture) that satellite L-space knots are braided satellites., 34 pages, 12 figures

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a "hyperbolic seiferter" or a "hyperbolic annular pair" produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots which have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted., Comment: To appear in Algebr. Geom. Topol

A non-trivial slope $r$ on a knot $K$ in $S^3$ is called a characterizing slope if whenever the result of $r$-surgery on a knot $K'$ is orientation preservingly homeomorphic to the result of $r$-surgery on $K$, then $K'$ is isotopic to $K$. Ni and Zhang ask: for any hyperbolic knot $K$, is a slope $r = p/q$ with $|p| + |q|$ sufficiently large a characterizing slope? In this article we answer this question in the negative by demonstrating that there is a hyperbolic knot $K$ in $S^3$ which has infinitely many non-characterizing slopes. As the simplest known example, the hyperbolic knot $8_6$ has no integral characterizing slopes., 13 pages, 7 figures

In [A. Deruelle, K. Miyazaki, K. Motegi, Networking Seifert surgeries on knots, Preprint], we have defined the Seifert Surgery Network consisting of all the integral Dehn surgeries on knots in S 3 yielding Seifert fiber spaces, where Seifert fiber spaces may have fibers of indices zero as a degenerate case. In this paper we focus on lens surgeries, i.e. Dehn surgeries on knots which yield lens spaces. J. Berge [J. Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript] gave a list of twelve infinite families of lens surgeries, which is conjectured to be a complete list of lens surgeries. We locate the Berge's lens surgeries in the Seifert Surgery Network, and show that in the network, they are “close” to Seifert surgeries on torus knots.

Let M be a small Seifert fiber space which has also a structure of surface bundle F × [0, 1]/{(x, 0) = (f(x), 1)} over the circle, where f: F → F is a monodromy map with non-empty fixed point set. A typical example of such a manifold appears as the result of 0-surgery on a torus knot. For each section in M, we have a ‘projection’ in F in a natural way. We give a condition assuring that the given section in M is hyperbolic in terms of the ‘projection’ in the fiber surface. By translating the result, we give a condition to obtain pseudo-Anosov automorphisms of once punctured surfaces from a periodic automorphism.

Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $\omega$ between $K$ and $c$. We establish the conjecture in case of $|\omega| \neq 1$, prove that $\{K_n\}$ contains at most three L-space knots if $\omega = 0$, and address the case where $|\omega| = 1$ under an additional hypothesis about Seifert surgeries. To that end, we characterize a twisting circle $c$ for which $\{ (K_n, r_n) \}$ contains at least ten Seifert surgeries. We also pose a few questions about the nature of twist families of L-space knots, their expressions as closures of positive (or negative) braids, and their wrapping about the twisting circle., Comment: In addition to addressing typos, V2 restructures, expands, and corrects parts of V1 and incorporates comments of referees. Accepted by Comm. Anal. Geom

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish., Comment: We found a gap in the proof of Lemma 3.1 in the last version (1501.01105v2). This lemma is replaced by Proposition 3.2 in [Kalfagianni and Tran; Knot cabling and the degree of the colored Jones polynomial (1501.01574v1)]. The title of the paper is also changed

We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yields an L-space knot. Consequently, any knot obtained by a non-trivial band sum cannot admit a finite surgery. For context, we exhibit two examples of non-trivial band sums of tight fibered knots producing prime knots: one is fibered but not tight, and the other is strongly quasipositive but not fibered., Comment: 8 pages, 6 figures

Let K be a knot in the 3-sphere S 3 , and a disk in S 3 meeting K transversely in the interior. For non-triviality we assume that | \ K| 2 over all isotopies of K in S 3 @ . Let K ,n( S 3 ) be a knot obtained from K by n twistings along the disk . If the original knot is unknotted in S 3 , we call K ,n a twisted knot. We describe for which pair (K,) and an integer n, the twisted knot K ,n is a torus knot, a satellite knot or a hyperbolic knot.

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3, 3,5)-pretzel knot belongs to both of the infinite families.

J. Dean introduced the notion of primitive/Seifert-fibered knots and constructions to explain how Dehn surgeries on knots produce small Seifert fibered manifolds. We determine non-hyperbolic, primitive/Seifert-fibered knots, and show that for such knots any integral, small Seifert fibered surgery arises from a primitive/Seifert-fibered construction. We also show that any 2-orbifold .

Let K be a knot in the 3-sphere S 3 and D ad isk inS 3 meeting K transversely more than once in the interior. For nontriviality we assume that |D ∩ K| 2 over all isotopies of K in S 3 − ∂D .L et KD,n (⊂ S 3 ) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and KD,n is a graph knot, then |n| 1o rK and D form a special pair which we call an “exceptional pair”. As a corollary, if (K, D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.  2004 Elsevier B.V. All rights reserved. MSC: 57M25

Let K be a knot in the 3-sphere S3, and D a disk in S3 meeting K transversely in the interior. For non-triviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n(⊂S3) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S3, we call KD,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot KD,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. Ait Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Jeff Weeks’ computer program SnapPea has been used widely in 3-manifold topology. This program computes hyperbolic structures after drillings and Dehn fillings on 3-manifolds, and it provides a variety of associated topological, geometric and arithmetic invariants. In our previous study about Seifert fibered Dehn surgeries on knots, we used SnapPea to investigate a relationship between closed geodesics in hyperbolic knot complements and Seifert fibers after Seifert fibered surgeries on them. We will explain how we used SnapPea in the study and propose some questions inspired by the computer experiments. These experiments were carried out in the joint work with Katura Miyazaki while we were preparing the paper [Miyazaki and Motegi, Comm. Anal. Geom., 7: 551–582].

We show that if a periodic knot K in the 3-sphere yields a Seifert fibered manifold by Dehn surgery, then the quotient of K by the group action generated by any periodic map of K is a torus knot, except for a special case. We also consider what Seifert fibered manifolds are obtained by Dehn surgery on periodic knots. If a non-torus, periodic knot yields a Seifert fibered manifold M , then the base space of M is the 2-sphere; and some pair of exceptional fibers in M has indices coprime provided that M contains at most three exceptional fibers.

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one., The final version, accepted for publication by Algebr. Geom. Topol

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m which is not -2 it contains a successive triple of Seifert surgeries (K, m), (K, m +1), (K, m +2) on a hyperbolic knot K, e.g. 17-, 18-, 19-surgeries on the (-2, 3, 7) pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifert-fibered construction, e.g. (-1)-surgery on the (3, -3, -3) pretzel knot., Comment: To appear in Bol. Soc. Mat. Mexicana

We obtain an infinite family of hyperbolic knots in a solid torus which admit half-integral, toroidal and annular surgeries. Among this family we find a knot with two toroidal and annular surgeries; one is integral and the other is half-integral, and their distance is 5. This example realizes the maximal distance between annular surgery slopes and toroidal ones, and that between annular surgery slopes.

Let K K be a knot in the 3 3 -sphere S 3 S^{3} , and D D a disc in S 3 S^{3} meeting K K transversely more than once in the interior. For non-triviality we assume that | K ∩ D | ≥ 2 \vert K \cap D \vert \ge 2 over all isotopy of K K . Let K n K_{n} ( ⊂ S 3 \subset S^{3} ) be a knot obtained from K K by cutting and n n -twisting along the disc D D (or equivalently, performing 1 / n 1/n -Dehn surgery on ∂ D \partial D ). Then we prove the following: (1) If K K is a trivial knot and K n K_{n} is a composite knot, then | n | ≤ 1 \vert n \vert \le 1 ; (2) if K K is a composite knot without locally knotted arc in S 3 − ∂ D S^{3} - \partial D and K n K_{n} is also a composite knot, then | n | ≤ 2 \vert n \vert \le 2 . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable 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., 23 pages, substantial revisions of the previously posted paper. To appear in Comm. Anal. Geom

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced in [9]. The Seifert surgery network is a 1‐dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted. 57M25; 57M50, 57N10 Dedicated to Sadayoshi Kojima on the occasion of his 60 th birthday

Throughout this paper we work in the smooth category and assume that all knots are oriented and consider two knots K1 and K2 to be equivalent if and only if there is an orientation preserving homeomorphism h:S3→S3 which carries K1 onto K2 so that their orientations match. We write K1 ≅ K2 if K1 and K2 are equivalent and – K denotes the knot obtained from K by inverting its orientation.

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume., Comment: 13 pages, 12 figures

Let V V be a standardly embedded solid torus in S 3 {S^3} with a meridian-preferred longitude pair ( μ , λ ) (\mu ,\lambda ) and K K a knot contained in V V . We assume that K K is unknotted in S 3 {S^3} . Let f n {f_n} be an orientation-preserving homeomorphism of V V which sends λ \lambda to λ + n μ \lambda + n\mu . Then we get a twisted knot K n = f n ( K ) {K_n} = {f_n}(K) in S 3 {S^3} . Primeness of twisted knots is discussed and we prove: A twisted knot K n {K_n} is prime if | n | > 5 |n| > 5 . Moreover, { K n } n ∈ Z {\{ {K_n}\} _{n \in Z}} contains at most five composite knots.

A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries., 13 pages. arXiv admin note: text overlap with arXiv:1202.4211

Let F be a compact, orientable surface with negative Euler characteristic, and let x 1 , ··· ,x n be n fixed but arbitrarily chosen points on int F , each of which has a (small) diskal neighborhood D i ⊂ F . Denote by S n (F) a subgroup of Diff (F) consisting of “sliding” maps f each of which satisfies ( 1 )f( { x 1 , ··· ,x n } ) ={ x 1 , ··· ,x }, f(D 1 ∪···∪ D ) = D 1 ∪···∪ D n and ( 2 )f is isotopic to the identity map on F .Then by restricting such automorphismsto F ˆ = F −int (D 1 ∪···∪ D n ) ,wehave automorphisms f ˆ : Fˆ → Fˆ ,whichform a subgroup S n (F) ˆ of Diff (F) ˆ . We give a Nielsen-Thurston classification of elements of S n (F) ˆ using braids in F × I which characterize the elements of S n (F) ˆ . 1. Introduction An automorphism (i.e., orientation preserving self diffeomorphism) of a compact, ori-entable surface with possibly non-empty boundary is said to be periodic if its some power isequal to the identity map, and is said to be reducible if it leaves an essential

Which slopes can or cannot appear as Seifert fibered slopes for hyperbolic knots in the 3-sphere S^3? It is conjectured that if r-surgery on a hyperbolic knot in S^3 yields a Seifert fiber space, then r is an integer. We show that for each integer n, there exists a tunnel number one, hyperbolic knot K_n in S^3 such that n-surgery on K_n produces a small Seifert fiber space., Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-16.abs.html