Tomotada Ohtsuki

On a universal perturbative invariant of 3-manifolds

Using finite type invariants (or Vassiliev invariants) of framed links and the Kirby calculus we construct an invariant of closed oriented three-dimensional manifolds with values in a graded Hopf algebra of certain kinds of 3-valent graphs (of Feynman diagrams). The degree 1 part of the invariant is essentially the Casson-Lescop-Walker invariant of 3-manifolds. A generalization for links in 3-manifolds is also given. The theory of this invariant can be regarded as a mathematically rigorous realization of part of Witten’s theory of quantum invariants in [33]. For a 3-manifold M, a compact Lie group G, and an integer k, Witten claimed that Zk(M, G) = s eJ _ 1kCS(A)23A

Invariants of three-manifolds derived from linking matrices of framed links

Introduction. In [16], R. Kirby and P. Melvin study invariants of 3manifolds τr(r>3) introduced by E. Witten [38], N. Reshetikhin and V.G. Turaev [31], and W.B.R. Lickorish [25, 26, 27] (see also [18]). In particular, Kirby and Melvin calculated τ3 and τ4 explicitly. Let M be a closed, oriented 3-manifold obtained from an (integral) framed link L. Then τ3(M) can be written as follows [16, §6].

Epimorphisms between 2-bridge link groups

We give a systematic construction of epimorphisms between 2‐bridge link groups. Moreover, we show that 2‐bridge links having such an epimorphism between their link groups are related by a map between the ambient spaces which only have a certain specific kind of singularity. We show applications of these epimorphisms to the character varieties for 2‐bridge links and 1 ‐dominating maps among 3‐ manifolds.

INVARIANTS OF COLORED LINKS

We define a new hierarchy of isotopy invariants of colored oriented links through oriented tangle diagrams. We prove the colored braid relation and the Markov trace property explicitly.

COLORED RIBBON HOPF ALGEBRAS AND UNIVERSAL INVARIANTS OF FRAMED LINKS

For colored representations of Uq(sl2), we give explicit formulas of universal R-matrices and construct universal invariants of framed links.

Quantum SU(3) Invariant of 3-Manifolds via Linear Skein Theory

The linear skein theory for the Kauffman bracket was introduced by Lickorish [11,12]. It gives an elementary construction of quantum SU(2) invariant of 3-manifolds. In this paper we prove basic properties of the linear skein theory for quantum SU(3) invariant. By using them we give an elementary construction of quantum SU(3) invariant of 3-manifolds and prove topological invariance of the invariant along the construction.

Invariants of 3-manifolds derived from universal invariants of framed links

Reshetikhin and Turaev [ 10 ] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group U q ( sl 2 )) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For U q ( sl 2 ) these two methods give different invariants of 3-manifolds.

The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant

Abstract For a Lie algebra g and its representation R, the quantum ( g , R) invariant of knots recovers from the Kontsevich invariant through the weight system derived from substitution of g and R into chord diagrams. We expect a similar property for invariants of 3-manifolds; for a Lie group G, the perturbative G invariant of 3-manifolds should recover from the universal perturbative invariant defined in [25] through the weight system derived from substitution of the Lie algebra of G. In this paper we give a rigorous proof of the recovery for G = SO (3).

The perturbative invariants of rational homology 3-spheres can be recovered from the LMO invariant

We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the Le-Murakami-Ohtsuki (LMO) invariant for any simple Lie algebra g ,t hat is, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in Le, Murakami and Ohtsuki [‘On a universal perturbative invariant of 3-manifolds’, Topology 37 (1998) 539–574]. Since the perturbative invariants dominate the quantum invariants of integral homology 3-spheres [K. Habiro, ‘On the quantum sl2 invariants of knots and integral homology spheres’, Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Topology Monographs 4 (Geometry and Topology Publications, Coventry, 2002) 161–181; K. Habiro, ‘A unified Witten– Reshetikhin–Turaev invariant for integral homology spheres’, Invent. Math. 171 (2008) 1–81; K. Habiro and T. T. Q. Le, in preparation], the LMO invariant dominates the quantum invariants of integral homology 3-spheres. In the late 1980s, Witten [35] proposed topological invariants of a closed 3-manifold M for a simple compact Lie group G, which is formally presented by a path integral whose Lagrangian is the Chern–Simons functional of G connections on M . There are two approaches to obtain mathematically rigorous information from a path integral: the operator formalism and the perturbative expansion. Motivated by the operator formalism of the Chern–Simons path integral, Reshetikhin and Turaev [33] gave the first rigorous mathematical construction of quantum invariants of 3-manifolds, and, after that, rigorous constructions of quantum invariants of 3-manifolds were obtained by various approaches. When M is obtained from S 3 by surgery along a framed knot K, the quantum G invariant τ G

Vanishing of 3-loop Jacobi diagrams of odd degree

We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop Vassiliev invariant can distinguish between a knot and its inverse.