Complexity for link complement states in Chern-Simons theory

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 M3. 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 Zk - 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 a general compact gauge group.