halacha - Is a wooden knife מקבל טומאה?

I know that פשוטי כלי עץ טהורים (wooden vessels that are flat [without a receptacle] cannot become טמא [ritually impure]). (כלים ב:א)

What about a wooden knife? It has a function, but doesn't have a בית קיבול (receptacle).
Are wooden knives מקבל טמא?

I'd prefer a halachic source (Rif, Rosh, Rambam, Shulchan Aruch et al) who discusses wooden knives directly, if possible.