On Vassiliev knot invariants induced from finite type 3-manifold invariants

Abstract

We prove that the knot invariant induced by a $\Bbb Z$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\Bbb Z$-homology 3-sphere invariant of order 5. This result agrees with a conjecture of Rozansky based on physical predictions about the asymptotic behavior of Witten's Chern-Simons path integral.

14 pages, amslatex, 9 figures not included. The compressed archive of the amslatex file and 9 postscript figure files can be obtained at ftp://math.columbia.edu/pub/lin/gl.tar.Z