A combinatorial view of Holant problems on higher domains

On the Boolean domain, there is a class of symmetric signatures called ``Fibonacci gates’’ for which a beautiful P-time combinatorial algorithm has been designed for the corresponding $\mathrm{Holant}$ problems. In this work, I give a combinatorial view for $\mathrm{Holant}(\mathcal{F})$ problems on a domain of size 3 where $\mathcal{F}$ is a set of arity 3 functions with inputs taking values on the domain of size 3 and the functions share some common properties. The combinatorial view can also be extended to the domain of size 4. Specifically, I extend the definition of “Fibonacci gates” to the domain of size 3 and the domain of size 4. Moreover, I give the corresponding combinatorial algorithms.