Applications of quantum invariants in low dimensional topology

In this short note we give lower bounds for the Heegaard genus of 3-manifolds using various TQFT in 2 + 1 dimensions. We also study the large k limit and the large G limit of our lower bounds, using a conjecture relating the various combinatorial and physical TQFTs. We also prove, assuming this conjecture, that the set of colored SU(N) polynomials of a framed knot in S3 distinguishes the knot from the unknot.