> I do not cut it. I dig out any shoots and as much root as I can as soon as they appear above ground.
That's a lot of digging!
Cutting vs digging shouldn't make much of a different in terms of the plant's response to damage (aggressive spreading). It'll spread when it gets hurt either way. Also, knotweed's root system goes down down 4-12 feet so usually removing the roots isn't feasible. Frequently digging it out or cutting it will hurt the plant but it's going to be a very slow and labor intensive way to eradicate it.
Yeah, it is a lot of digging. I already dug out all of the roots I could, which meant digging a waist deep hole and taking half a ton of the root stuff to the recycling center (to be burned). But now I'm dealing with the remaining roots trying to spread.
When I dig out the shoots, I always try to get more than this year's growth, and hope that laws of thermodynamics will eventually prevail.
The damn thing is almost dead by now. I get a dozen shoots a week or so.
For the other patch, I did just what you say, cut, herbicide and then covered it.