Structure in the classical knot concordance group

We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by von Neumann signatures.     

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).     

Grope cobordism of classical knots

Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants.The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobordism. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at h=2, Casson-Gordon invariants at h=3, and for h=4 one finds the new von Neumann signatures of a knot.     

Homology cobordism and classical knot invariants

In this paper, we define and investigate -homology cobordism invariants of -homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the -homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given -homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot. Received: May 7, 2001     

Inequivalent surface-knots with the same knot quandle

We have a knot quandle and a fundamental class as invariants for a surface-knot. These invariants can be defined for a classical knot in a similar way, and it is known that the pair of them is a complete invariant for classical knots. In surface-knot theory the situation is different: There exist arbitrarily many inequivalent surface-knots of genus g with the same knot quandle, and there exist two inequivalent surface-knots of genus g with the same knot quandle and with the same fundamental class.     

The Thurston norm, fibered manifolds and twisted Alexander polynomials

Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal ‘complexity’ of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita-Terasaka knot which have trivial Alexander polynomial.We also give obstructions to fibering 3-manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some of these it was unknown whether or not they are fibered. Our work in particular extends the fibering obstructions of Cha to the case of closed manifolds.     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.     

On some classes of hyperbolic knots with the 3-sphere surgery

For the distance of (1,1)-splittings of a knot in a closed orientable 3-manifold, it is an important problem whether a (1,1)-knot can admit (1,1)-splittings of different distances. In this paper, we give one-parameter families of hyperbolic (1,1)-knots such that each (1,1)-knot admits a Dehn surgery yielding the 3-sphere. It is remarkable that such knots are the first concrete examples each of whose (1,1)-splittings is of distance three.     

Counterexamples to Kauffman's conjectures on slice knots

In the late 1960s Jerome Levine classified the odd high-dimensional knot concordance groups in terms of a linking matrix associated to an arbitrary bounding manifold for the knot. His proof fails for classical knots in S³$S^3$. Yet this philosophy has remained the only known strategy for understanding the classical knot concordance group. We show that this strategy is fundamentally flawed. Specifically, in 1982, in support of Levine's philosophy, Louis Kauffman conjectured that if a knot in S³$S^3$ is a slice knot then on any Seifert surface for that knot there exists a homologically essential simple closed curve of self-linking zero which is itself a slice knot, or at least has Arf invariant zero. Since that time, considerable evidence has been amassed in support of this conjecture. In particular, many invariants that obstruct a knot from being a slice knot have been explicitly expressed in terms of invariants of such curves on its Seifert surface. We give counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice knots that admit (unique minimal genus) Seifert surfaces on which every homologically essential simple closed curve of self-linking zero has non-zero Arf invariant and non-zero signatures.     

On equivariant slice knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots- Naik's and Choi-Ko-Song's improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots - but are not equivariantly slice.

     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

Intersection Alexander polynomials

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander polynomials that are defined for non-locally-flat knots. We also investigate the relations between the intersection Alexander polynomials of a knot and the intersection and classical Alexander polynomials of the link knots around the singular strata. To facilitate some of these investigations, we introduce spectral sequences for the computation of the intersection homology of certain stratified bundles.     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

Maximal Thurston-Bennequin numbers of alternating links

We show that the upper bound of the maximal Thurston-Bennequin number for an oriented alternating link given by the Kauffman polynomial is sharp. As an application, we confirm a question of Ferrand. We also give a formula of the maximal Thurston-Bennequin number for all two-bridge links. Finally, we introduce knot concordance invariants derived from the Thurston-Bennequin number and the Maslov number of a Legendrian knot.     

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.     

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ?rhus invariant of knots. Received February 14, 2000 / Published online October 11, 2000     

Knot exteriors with all possible meridional essential surfaces

We show the existence of infinitely many knots where each exterior contains meridional essential surfaces of independently unbounded genus and number of boundary components. In particular, we construct examples of knot exteriors each of which has all possible compact orientable surfaces embedded as meridional essential surfaces. From these results, we also prove the existence of a hyperbolic knot exterior, not necessarily of a knot in the 3-sphere, for which there are meridional essential surfaces of simultaneously unbounded genus and number of boundary components.     

Note on the Casson-Gordon invariant of a satellite knot

In this paper we establish a relation between an appropriate version of the Casson-Gordon invariant of a satellite knot and those of its orbit and companion. We note that in some cases the contribution from, the companion falls. This gives a way to construct algebraically but not smoothly slice knots. This article was processed by the author using theLaT_EX style filecljouri from Springer-Verlag.     

Isolated exceptional Dehn surgeries on hyperbolic knots

For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic manifolds. Such exceptional surgeries are classified into four types, lens space surgery, small Seifert fibered surgery, toroidal surgery and reducing surgery, according to the resulting manifolds. For each of the three types except reducing surgery, we give infinitely many hyperbolic knots with integral exceptional Dehn surgeries of the given type, whose adjacent integral surgeries are not exceptional.     

Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.