Hi Jennifer,
There was a big aggressive patch of Japanese knotweed at the back of my home. My neighbor and I were both taking measures and tried to avoid chemicals. One year my landscaper tried to smother it with mounds of wood chips - it helped a little, but as you know little bits of it pop up and take off quickly.
This is how our saga ended.
At the end of the season a few years ago I tossed out into the back of my yard a few Long Island Cheese pumpkins - just heaved them out there where they disintegrated. We had gotten too many from our CSA. The next year in the fall we saw a huge pumpkin vine. Somehow it created a hostile environment for the Japanese knotweed. The only "downside" is that the pumpkin vine went everywhere and volunteers each year, but I would so much rather deal with an "invasive" pumpkin vine than the dreaded knotweed.
Good Luck!
