We generalize Ngs two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Ac
tually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ngs three-variable knot contact homology. Our main tool is Lins generalization of the Markov theorem for braids in $S^3$ to braids in $S^1 times S^2$. We conjecture that our framed cord algebras are always finitely generated for non-local knots.
We show that braidings of the metaplectic anyons $X_epsilon$ in $SO(3)_2=SU(2)_4$ with their total charge equal to the metaplectic mode $Y$ supplemented with measurements of the total charge of two metaplectic anyons are universal for quantum computa
tion. We conjecture that similar universal computing models can be constructed for all metaplectic anyon systems $SO(p)_2$ for any odd prime $pgeq 5$. In order to prove universality, we find new conceptually appealing universal gate sets for qutrits and qupits.
Harnessing non-abelian statistics of anyons to perform quantum computational tasks is getting closer to reality. While the existence of universal anyons by braiding alone such as the Fibonacci anyon is theoretically a possibility, accessible anyons w
ith current technology all belong to a class that is called weakly integral---anyons whose squared quantum dimensions are integers. We analyze the computational power of the first non-abelian anyon system with only integral quantum dimensions---$D(S_3)$, the quantum double of $S_3$. Since all anyons in $D(S_3)$ have finite images of braid group representations, they cannot be universal for quantum computation by braiding alone. Based on our knowledge of the images of the braid group representations, we set up three qutrit computational models. Supplementing braidings with some measurements and ancillary states, we find a universal gate set for each model.
