We introduce the concept of "claspers," which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called "C k -equivalence," which is generated by surgery operations of a certain kind called "C k -moves". We prove that two knots in the 3-sphere are C k+1 -equivalent if and only if they have equal values of Vassiliev-Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev-Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3-dimensional topology. AMS Classification numbers Primary: 57M25Secondary: 57M05, 18D10

Abstract. We construct an invariant J M of integral homology spheres M with values in a completion Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU (2) Witten-ReshetikhinTuraev invariant τ ζ (M ) of M at ζ. Thus J M unifies all the SU (2) WittenReshetikhin-Turaev invariants of M . As a consequence, τ ζ (M ) is an algebraic integer. Moreover, it follows that τ ζ (M ) as a function on ζ behaves like an "analytic function" defined on the set of roots of unity. That is, the τ ζ (M ) for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τ ζ (M ) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.

Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prove some properties of this functorial LMO invariant, including its universality among rational finite-type invariants of Lagrangian cobordisms. Finally, we apply the LMO functor to the study of homology cylinders from the point of view of their finite-type invariants. 57M27; 57M25 IntroductionThe Kontsevich integral is an invariant of links in S 3 , the standard 3-sphere. In their papers [23; 24], Le and Murakami extended this invariant to a functor from the category of tangles in the standard cube OE 1; 1 3 to the category of Jacobi diagrams based on 1-manifolds. One of the main features of the Kontsevich integral is its universality among rational-valued finite-type invariants of tangles (in the Goussarov-Vassiliev sense).Le, Murakami and Ohtsuki constructed in [26] an invariant of closed oriented 3-manifolds, which is called the Le-Murakami-Ohtsuki invariant. The LMO invariant is defined from the Kontsevich integral via surgery presentations of 3-manifolds in S 3 . For rational homology 3-spheres, the LMO invariant is universal among rational-valued finite-type invariants (in the Ohtsuki sense). Later, Murakami and Ohtsuki extended in [34] the LMO invariant to an invariant of 3-manifolds with boundary, which satisfies modified axioms of TQFT. More recently, Le and the first author constructed from the LMO invariant a functor from a certain category of 3-dimensional cobordisms to a certain category of modules [6]. Let us recall that each of those two constructions [34; 6] (ii) Unify the extended Kontsevich integral and the LMO invariant into a single invariant Z.M; G/ of couples .M; G/, where M is a closed oriented 3-manifold and G M is an embedded framed trivalent graph.Then, a compact oriented 3-manifold with boundary is obtained from each couple .M; G/ by cutting a regular neighborhood N.G/ of G in M . If the connected components of G were split into two parts, say the "top" part G C and the "bottom" part G , then M n N.G/ can be regarded as a cobordism between closed surfaces, namely from @N.G C / to @N.G /. Finally, the LMO invariant of the cobordism M n N.G/ is defined in [34; 6] to be the Kontsevich-LMO invariant Z.M; G/ of the couple .M; G/.In this paper, we propose an alternative solution to the problem of extending the LMO invariant to 3-manifolds with boundary. In contrast with the previous two constructions [34; 6], we prefer to work with cobordisms between once-punctured surfaces. This technical choice has two advantages: On the one hand, it avoids extending the Kontsevich integral to trivalent graphs in S 3 ; on the other hand, it allows us to work with monoidal categories, and to construct tensor-preserving functors.Moreover, we normalize the Kontsevich-LMO invariant Z to obtain ...

A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of "braided Hopf algebra action" on the set of bottom tangles.Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H , we define a braided functor J from B to the category Mod H of left H -modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in Mod H .Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants. 57M27; 57M25, 18D10 IntroductionThe notion of category of tangles (see Yetter [84] and Turaev [80]) plays a crucial role in the study of the quantum link invariants. One can define most quantum link invariants as braided functors from the category of (possibly colored) framed, oriented tangles to other braided categories defined algebraically. An important class of such functorial tangle invariants is introduced by Reshetikhin and Turaev [74]: Given a ribbon Hopf algebra H over a field k , there is a canonically defined functor F W T H ! Mod

We will announce some results on the values of quantum sl 2 invariants of knots and integral homology spheres. Lawrence's universal sl 2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl 2 . This implies an integrality result on the colored Jones polynomials of a knot. We define an invariant of integral homology spheres with values in a completion of the Laurent polynomial ring of one variable over the integers which specializes at roots of unity to the Witten-Reshetikhin-Turaev invariants. The definition of our invariant provides a new definition of Witten-Reshetikhin-Turaev invariant of integral homology spheres.