I can make the argument that swords are a special case but I think you are right; I am falling into the trap of sphere of influence must equal domain, which is not what we are going for.
So I would say Weapons as a domain. Han-Sui, Falan, Treetop, Mouser
I will work on it. You can also consider if you want to then make Martial/Combat= Unarmed Combat instead.