On the group-like behaviour of the Le-Murakami-Ohtsuki invariant

The Aarhus integral of rational homology 3-spheres III: The Relation with the Le-Murakami-Ohtsuki Invariant

Continuing the work started in Part I and II of this series (see q-alg/9706004 and math.QA/9801049), we prove the relationship between the Aarhus integral and the invariant $Ω$ (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in q-alg/9512002. The basic reason for the relationship is that both constructions afford an interpretation as "integrated holonomies". In the case of the Aarhus integral, this interpretation was the basis for everything we did in Parts I and II. The main tool we used there was "formal Gaussian integration". For the case of the LMO invariant, we de…

The Casson-Walker-Lescop Invariant as a Quantum 3-manifold Invariant

We give a direct computational proof that the degree 1 part of the Le-Murakami-Ohtsuki invariant of a closed oriented 3-manifold M is determined by the Casson-Walker-Lescop invariant. Moreover, if the first Betti number of M is equal to 2, the Le-Murakami-Ohtsuki invariant is determined by the Lescop generalization of the Casson-Walker invariant.

闭3-流形的Le-Murakami-Ohtsuki不变量综述

A survey on the Le-Murakami-Ohtsuki invariant for closed 3-manifolds

本文综述闭3-流形LMO不变量的原始构造方法，引入Kirby结构、pre-LMO结构等组合工具，结合Kontsevich积分得到两族乘法不变量，阐述LMO不变量的导出过程及相关唯一性结果，并补充了未被文献明确讨论的一族不变量的性质。

On Ohtsuki's invariants of integral homology 3-spheres, I

An attempt is made to conceptualize the derivation as well as to facilitate the computation of Ohtsuki's rational invariants $λ_n$ of integral homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants. Several interesting consequences will follow from our computation of $λ_2$. One of them says that $λ_2$ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that $λ_{1}$ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on kno…

Witten's Invariants of Rational Homology Spheres at Prime Values of $K$ and Trivial Connection Contribution

We establish a relation between the coefficients of asymptotic expansion of trivial connection contribution to Witten's invariant of rational homology spheres and the invariants that T.~Ohtsuki extracted from Witten's invariant at prime values of $K$. We also rederive the properties of prime $K$ invariants discovered by H.~Murakami and T.~Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki's invariants are of finite type. The relation between Ohtsuki…

The Aarhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S^3

Path integrals don't really exist, but it is very useful to dream that they do exist, and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S^3 don't really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology sph…