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This is a survey of the impact of Thurston's work on knot theory, laying emphasis on the two characteristic features, rigidity and flexibility, of 3-dimensional hyperbolic structures. We also lay emphasis on the role of the classical invariants, the Alexander polynomial and the homology of finite branched/unbranched coverings., Comment: 97 pages, 12 figures

This paper describes a Dehn surgery approach to generating asymmetric hyperbolic manifolds with two distinct lens space fillings. Such manifolds were first identified in work of Dunfield-Hoffman-Licata as the result of a computer search of the SnapPy census, but the current work establishes a topological framework for constructing vastly many more such examples. We introduce the notion of a ``jointly primitive'' presentation of a knot and show that a refined version of this condition ``longitudinally jointly primitive'' is equivalent to being surgery dual to a $(1,2)$--knot in a lens space. This generalizes Berge's equivalence between having a doubly primitive presentation and being surgery dual to a $(1,1)$--knot in a lens space. Through surgery descriptions on a seven-component link in $S^3$, we provide several explicit multi-parameter infinite families of knots in lens spaces with longitudinal jointly primitive presentations and observe among them all the examples previously seen in Dunfield-Hoffman-Licata., Comment: 33 pages, 16 figures, and ancillary files

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order >2., Comment: 36 pages, 14 figures

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also cusp shape. We demonstrate explicit solutions for $T(2,N)$ torus knots, $J(N,M)$ knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspect of the knot diagram., Comment: 55 pages, 31 figures

For a compact, orientable, irreducible 3-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, the Thurston norm of a class in the second homology of the filled manifold plus the so-called winding norm of the class will be equal to the Thurston norm of the corresponding class in the second homology of the unfilled manifold. This generalizes a result of Sela and is used to answer a question of Baker-Motegi concerning the Seifert genus of knots obtained by twisting a given initial knot along an unknot which links it.

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. We show that for every handlebody-knot pair associated to a genus two "prime bi-knotted" surface, one is irreducible, and the other is reducible. Furthermore, for given two genus two handlebody-knots $H_1$ and $H_2$ satisfying certain conditions, we will construct a "prime bi-knotted" surface whose associated handlebody-knot pair coincides with $H_1$ and $H_2$., Comment: 11 pages, 14 figures

We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections., Comment: 17 pages, 10 figures

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,v_{oct}]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants., Comment: 8 pages, 4 figures

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots., Comment: 8 pages, 2 figures. arXiv admin note: text overlap with arXiv:1411.7915. substantial text overlap with arXiv:1411.7915

Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit., Comment: 17 pages, 12 figures. Formerly part of arXiv:1411.7915. V2: minor corrections in sections 2 and 4; clarified and expanded exposition throughout

We show that every knot is one crossing change away from a knot of arbitrarily high bridge number and arbitrarily high bridge distance., Comment: Accepted by American Mathematical Monthly. New version incorporates referee comments

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios., Comment: 26 pages, 12 figures. V2: Updated references, removed hypothesis from Theorem 1.4, added Corollary 1.11, and made other minor wording changes. V3: Discussion on weaving knots has been moved into another paper. V4: Added Figures 10 and 12, minor wording changes

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.

Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 1995. We provide infinite families of closed braids for which our conditions hold., Comment: To appear in Asian Journal of Matematics. 18 pages, 12 figures

We show that the cusp volume of a hyperbolic alternating knot can be bounded above and below in terms of the twist number of an alternating diagram of the knot. This leads to diagrammatic estimates on lengths of slopes, and has some applications to Dehn surgery. Another consequence is that there is a universal lower bound on the cusp density of hyperbolic alternating knots., Comment: 21 pages, 8 figures; v4: revised final version, with corrected constants throughout the paper; to appear in Geometry & Topology

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold., Comment: This is the final (accepted) version of this paper

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311., Comment: 23 pages, 19 figures

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram., Comment: 17 pages, 11 figures

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: (1) The exterior of L contains an essential meridional sphere. (2) L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus. (3) L is obtained by banding from another link L' that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1., Comment: 27 pages, 13 figures

We complete the classification of hyperbolic pretzel knots admitting Seifert fibered surgeries. This is the final step in understanding all exceptional surgeries on hyperbolic pretzel knots. We also present results toward similar classifications for non-pretzel Montesinos knots of length three., Comment: 42 pages, 27 figures; Corrected typos, added reference, general incorporation of referee comments, including a significant addition to the proofs in Section 5. To appear in Algebraic and Geometric Topology

Riley "defined" the Heckoid groups for 2-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from 2-bridge link groups onto Heckoid groups. In this paper, we make Riley's definition explicit, and give a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's construction., Comment: 26 pages, 7 figures; Remark 2.1 added; updated version, incorporating the referee's comments; to appear in Hiroshima Mathematical Journal

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S', each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S \cup S' such that M contains a c-component tangle T with a bridge surface Sigma of genus g that separates the boundary of M into S and S', |T \cap Sigma|=2b and T intersects S and S' exactly in their marked points, and Sigma has distance at least n., Comment: 17 pages, 13 figures; v2 clarifying revisions made based on referee's comments

The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted torus knots, we consider the twisted torus knots to partially answer the question of which primitive/Seifert knots are fibered. A braid computation shows that a particular family of twisted torus knots is fibered, and that computation is then used to generalize the results of a previous paper by the author., Comment: 13 pages, 6 figures

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

An alternative method is described for determining the hyperbolic structure on a link complement, and some of its elementary consequences are examined. The method is particularly suited to alternating links., Comment: 32 pages, 16 figures, to appear in Algebraic & Geometric Topology

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized augmented links., Comment: 33 pages, 12 figures

In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or $\lambda_1$. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on $\lambda_1$. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded., Comment: 16 pages, 7 figures

We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based only on a diagram. The results are obtained by finding geometric information on generalized augmentations of these links., Comment: 16 pages, 6 figures. v2: Article has been split into two. New version contains only hyperbolic geometry results. Seifert fibered and toroidal results are now in the paper entitled "On multiply twisted knots and links which are Seifert fibered or toroidal", arXiv:0906.4575. Results similar to v1, but paper has been rewritten, with new title.

The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]. This paper is correct, but contains less information than the new one. The topological classification of knots that are closed braids is shown to lead to a classification theorem for transversal knots that are closed 3-braids. A list is given of all low crossing examples of transversally non-simple knots that are closed 3-braids., Comment: The contents of this 6-page paper have been subsumed into the 13-page paper, "A note on closed 3-braids", arXiv:0802.1072 [math.GT]

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4, and we find infinitely many families with meridian length approaching 4 from below. Finally, we present an example to show that, in contrast to the case of the regular augmented link, longitude lengths of these knots cannot be determined by a function of the number of twist regions alone., Comment: v2: 20 pages, 13 figures. Simplified proofs of main results and added two sections, one giving examples of knots with meridian lengths approaching the upper bound of 4, and one showing that there are no bounds on longitude length in terms of twist number. Updated the title to reflect these changes. To appear Comm. Anal. Geom

Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2\pi. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials., Comment: This version contains corrections to Section 4. Published in Journal of Differential Geometry

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements., Comment: 42 pages, 23 figures. Slightly expanded exposition and references

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., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-16.abs.html

For a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if M is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds., Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-21.abs.html

We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link., Comment: 28 pages, 15 figures. Minor rewording and organizational changes; also added theorem giving a lower bound on the genus of these links

These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and contact structures on Euclidean 3-space. I discuss my recent work with William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the latter we prove that there are distinct transversal knot types in contact 3-space having the same topological knot type and the same Bennequin invariant., Comment: 10 pages, 5 figures

Choose any oriented link type X and closed braid representatives X[+], X[-] of X, where X[-] has minimal braid index among all closed braid representatives of X. The main result of this paper is a `Markov theorem without stabilization'. It asserts that there is a complexity function and a finite set of `templates' such that (possibly after initial complexity-reducing modifications in the choice of X[+] and X[-]which replace them with closed braids X[+]', X[-]') there is a sequence of closed braid representatives X[+]' = X^1->X^2->...->X^r = X[-]' such that each passage X^i->X^i+1 is strictly complexity reducing and non-increasing on braid index. The templates which define the passages X^i->X^i+1 include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index m>= 4 a finite set T(m) of new ones. The number of templates in T(m) is a non-decreasing function of m. We give examples of members of T(m), m>= 4, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper., Comment: This is the version published by Geometry & Topology on 27 April 2006; part II (arXiv:math/0310280) is also published in GT volume 10

This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published explanations of SnapPea's algorithms for triangulating a link complement efficiently and for converging quickly to the hyperbolic structure while avoiding singularities in the parameter space., Comment: To appear in the Handbook of Knot Theory. 26 pages, 22 figures

We give a summary of known results on the maximal distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing a surface of non-negative Euler characteristic that is either essential or Heegaard., Comment: 23 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper10.abs.html