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16
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Claspers and finite type invariants of links
, 2000
"... 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 “Ck–equivalence,” which is generated by surgery operatio ..."
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Cited by 26 (1 self)
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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 “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+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.
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 17 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM 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) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 14 (2 self)
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We construct an invariant JM 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) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(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, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.
Towards an Algebraic Characterization of 3dimensional Cobordisms. ArXiv: math.GT/0106253
"... (To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dim ..."
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Cited by 7 (0 self)
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(To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3dimensional relative cobordisms, the second is a
On the center of the small quantum group
 J. of Algebra
"... Abstract. Using the quantum Fourier transform F [LM], we describe the block decomposition and multiplicative structure of a subalgebra ˜ Z+ F ( ˜ Z) of the center of the small quantum group U fin q (g) at a root of unity. It contains the known subalgebra ˜ Z [BG], which is isomorphic to the algebra ..."
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Cited by 6 (2 self)
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Abstract. Using the quantum Fourier transform F [LM], we describe the block decomposition and multiplicative structure of a subalgebra ˜ Z+ F ( ˜ Z) of the center of the small quantum group U fin q (g) at a root of unity. It contains the known subalgebra ˜ Z [BG], which is isomorphic to the algebra of characters of finite dimensional U fin q (g)modules. We prove that the intersection ˜ Z ∩ F ( ˜ Z) coincides with the annihilator of the radical of ˜ Z. Applying representationtheoretical methods, we show that ˜ Z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over U fin q (g). In particular this leads to the conclusion that the center of U fin q (g) coincides with ˜ Z+ F ( ˜ Z) in the case g = sl2.
On the connectivity of cobordisms and halfprojective TQFT’s
 Comm. Math. Phys
, 1998
"... Abstract: We consider a generalization of the axioms of a TQFT, so called halfprojective TQFT’s, where we inserted an anomaly, x µ0, in the composition law. Here µ0 is a coboundary (in a group cohomological sense) on the cobordism categories with nonnegative, integer values. The element x of the r ..."
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Cited by 6 (5 self)
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Abstract: We consider a generalization of the axioms of a TQFT, so called halfprojective TQFT’s, where we inserted an anomaly, x µ0, in the composition law. Here µ0 is a coboundary (in a group cohomological sense) on the cobordism categories with nonnegative, integer values. The element x of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be zero. This modification makes it possible to extend quantuminvariants, which vanish on S 1 ×S 2, to nontrivial TQFT’s. Note, that a TQFT in the ordinary sense of Atiyah with this property has to be trivial all together. We organize our discussions such that the notion of a halfprojective TQFT is extracted as the only possible generalization under a few very natural assumptions. Based on separate work with Lyubashenko on connected TQFT’s, we construct a large class of halfprojective TQFT’s with x = 0. Their invariants all vanish on S 1 × S 2, and they coincide with the Hennings invariant for nonsemisimple Hopf algebras and, more generally, with the Lyubashenko invariant for nonsemisimple categories. We also develop a few topological tools that allow us to determine the cocycle µ0 and find numbers, ̺(M), such that the linear map associated to a cobordism, M, is of the form x ̺(M) fM. They are concerned with connectivity properties of cobordisms, as for example maximal nonseparating surfaces. We introduce
Invariants of spin threemanifolds from ChernSimons theory and finitedimensional Hopf algebras
 Adv. Math
"... Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations correspon ..."
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Cited by 6 (2 self)
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Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the halfinteger level ChernSimons theories conjectured to give spin TQFTs by Dijkgraaf and Witten [10]. In particular, the spin invariants constructed by Kirby and Melvin [21] are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finitedimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg’s noninvolutory invariant of framed manifolds associated to that Hopf algebra.
Bottom tangles and universal invariants
, 2006
"... 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 ..."
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Cited by 5 (2 self)
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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 ModH 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 ModH. 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.
Threedimensional 2framed TQFTs and surgery
"... Abstract. The notion of 2framed threemanifolds is defined. The category of 2framed cobordisms is described, and used to define a 2framed threedimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2f ..."
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Abstract. The notion of 2framed threemanifolds is defined. The category of 2framed cobordisms is described, and used to define a 2framed threedimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2framed threedimensional TQFT. These data and relations are expressed in the language of surgery.