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A knot $\kappa$ in $S^3$ is persistently foliar if, for each non-trivial boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a co-oriented taut foliation in every manifold obtained by non-trivial Dehn surgery on that knot. We show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is fibered or at least one summand is nontorus alternating or Montesinos is persistently foliar. We note that, in constructing foliations in the complements of fibered summands, we build branched surfaces whose complementary regions agree with those of Gabai's product disk decompositions, except for the one containing the boundary of the knot complement. It is this boundary region which provides for persistence., Comment: 37 pages, 30 figures. Added description of canonical meridian and two new figures. Improved exposition and stronger statement of results. To be published in Algebraic and Geometric Topology

A twisted torus knot is obtained from a torus knot by adding a number of full twists to some adjacent strands of the torus knot. In this paper, we show that if a twisted torus knot is a torus knot, then the number of added full twists is generically at most two in absolute value. We also show that this bound is the best possible by classifying twisted torus knots for which the upper bound is attained. 57N10

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot. 57M27

By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgeries are discussed., 27 pages; a few typos corrected; some corollaries removed and replaced by corollary about lens space surgeries

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon., Comment: Combines and expands arXiv:0908.2795 and arXiv:0901.2319 into the version published in Geometry and Topology

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots., 37 pages, 4 figures, improved exposition

We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms., 21 pages, 11 figures. Some errors in References are revised in version 2

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then either \begin{itemize} \item $d(P)\leq 2-\chi(Q-K)$, or \item K can be isotoped to be disjoint from Q so that after the isotopy Q is a Heegaard surface for the knot exterior and is isotopic to a possibly stabilized copy of P. \end{itemize}, Comment: References added, exposition slightly improved, 17 pages, 3 figures

Our main theorem is that, if [math] is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre [math] and pseudo-Anosov monodromy, then the lift of the inclusion of [math] in [math] to universal covers extends to a continuous map of [math] to [math] , where [math] . The restriction to [math] maps onto [math] and gives an example of an equivariant [math] –filling Peano curve. After proving the main theorem, we discuss the case of the figure-eight knot complement, which provides evidence for the conjecture that the theorem extends to the case when [math] is a once-punctured hyperbolic surface.

In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z semi-direct product Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals (t-2)(t^{-1}-2) but also contains information about the metabelian cover of the knot complement (since there are many non-slice knots with this Alexander polynomial). Erratum (attached): In Figure 1.5 we gave an incorrect example for Theorem 1.3. We present a correct example., This is the version published by GT on 4 November 2005 and includes the erratum published on 18 October 2006

J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin., Comment: 14 pages, 4 figures

A nontrivial knot that can be drawn with only two relative maxima in the vertical direction is called a 2-bridge knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking, a lamination in a manifold M is a foliation of M, except that it can have a nonempty open complement in M, and very loosely speaking, the lamination is essential if each leaf of it L is incompressible, i.e. inclusion of L into M induces an injective homomorphism from 1(L) into 1(M). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a nontorus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by R 3 , and have innite fundamental group. So Property P is true for non-torus 2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or a counterexample to the Poincare conjecture. We use general techniques which are not specic to 2-bridge knots to nd and explicitly construct these laminations. In trying to understand 3-manifolds with the hope of eventually classifying them, as with 2-manifolds, one approach that has turned out to be fruitful is to study objects of codimension one in them, more specically, incompressible surfaces, Reebless foliations, and essential laminations. For a 3-manifoldM containing an incompressible surface, with some extra hypotheses, Waldhausen proved: The universal cover of M is R 3 , the topology of M is determined by its fundamental group, and homotopic homeomorphisms of M are isotopic. Similar theorems for manifolds containing Reebless foliations were proven by Haefliger, Novikov, Palmeira, Rosenberg, and others. The essential lamination was developed in the 1980’s as a generalization of the incompressible surface and the Reebless foliation, which themselves qualify as essential laminations. In fact they are just extreme cases of essential laminations: At one end we have compact properly embedded surfaces

In this paper we consider Heegaard splittings of 3-manifolds. By using Gabai’s concept of thin position on the 1-skeleton of some polyhedral decomposition, together with CassonGordon’s concept of strong irreducibility, we prove the Main Theorem (4.0). This theorem will allow us to classify the Heegaard splittings of manifolds whose polyhedral decompositions are particularily nice, which we demonstrate via examples. Specically, we use it to classify Heegaard splittings of several hyperbolic spaces, including the gure-8 knot complement (Example 6.4) and the genus 2 case of the 52-knot complement (Example 6.7).

Every knot, K, in *S3 has associated to it an equivalence class of matrices based on *S-equivalenc e of Seifert matrices. When the knot is altered by changing a crossing, the ^-equivalence class of the new knot is related to that of the original knot in a very specific way. This change in the Seifert matrices can be studied without regard to the underlying geometric situation, leading to a theory of algebraic crossing changes. Thus, the algebraic unknotting number may be defined as the smallest number of these algebraic crossing changes necessary to convert a Seifert matrix for the knot into a matrix for the unknot. A straightforwar d test of some well-known knot invariants will reveal that the algebraic unknotting number is one.

This paper uses some ideas from 3-dimensional topology to study knots in S4 . We show that the Poincare conjecture implies the existence of a non-fibered knot whose complement fibers homotopically. In a different direction, we show that Gromov's norm is an obstruction to a knot having a Seifert surface made out of Seifert fibered spaces, and hence to being ribbon. We also prove that any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold, so that every knot in S4 has a hyperbolic Seifert surface. One of the reasons that the study of knots in the 4-sphere has a special character is that the Seifert surfaces that such knots bound are 3-dimensional. Hence the peculiar nature of the topology of 3manifolds can lead to interesting behavior of 2-knots. In this paper we give several examples of this principle. The first example is to show that the 3-dimensional Poincare conjecture implies the existence of non-fibered (topological) knots in *S 4 whose exteriors are homotopy equivalent to the exterior of a fibered knot. (Similar phenomena have been noticed by J. Hillman and C. B. Thomas [12, 13] and S. Weinberger [32].) The second instance is to see how the existence of a "geometric structure" on a Seifert surface influences topological properties of the knot. Restrictions on the possible geometric structures are obtained via "Gromov's norm" of a 2-knot, defined below. We show that a knot with non-zero norm cannot have a Seifert surface which is a connected sum of Seifert-fiber ed 3-manifolds. In particular, the norm is seen to be an obstruction to a knot in S4 being ribbon. A similar obstruction has been found by Bruce Trace [30]. In contrast, we will show that any knot has a Seifert surface which is a hyperbolic manifold. This follows from Theorem 2.6, which states that any 3-manifold has an invertible homology cobordism to a hyperbolic manifold. A cobordism W from M to N is called invertible if there is another cobordism W, so that W U N W = M x I. Without the requirement that the homology cobordism be invertible, this theorem is due to R. Myers [23].