Restricted holant dichotomy on domain sizes 3 and 4

${\mathrm{Holant}}^{⁎}\left(f\right)$ denotes a class of counting problems specified by a constraint function f. We prove complexity dichotomy theorems for ${\mathrm{Holant}}^{⁎}\left(f\right)$ in two settings: (1) f is any symmetric arity-3 real-valued function on input of domain size 3. (2) f is any symmetric arity-3 $\{0,1\}$-valued function on input of domain size 4.