  We study notions of complexity for link complement states in Chern Simons
theory with compact gauge group $G$. Such states are obtained by the Euclidean
path integral on the complement of $n$-component links inside a 3-manifold
$M_3$. For the Abelian theory at level $k$ we find that a natural set of
fundamental gates exists and one can identify the complexity as differences of
linking numbers modulo $k$. Such linking numbers can be viewed as coordinates
which embeds all link complement states into $\mathbb{Z}_k ^{\otimes n(n-1)/2}$
and the complexity is identified as the distance with respect to a particular
norm. For non-Abelian Chern Simons theories, the situation is much more
complicated. We focus here on torus link states and show that the problem can
be reduced to defining complexity for a single knot complement state. We
suggest a systematic way to choose a set of minimal universal generators for
single knot complement states and then evaluate the complexity using such
generators. A detailed illustration is shown for $SU(2)_k$ Chern Simons theory
and the results can be extended to general compact gauge group.
