I helped my aunt eradicate some. It takes years and this is the method I used. You can also use triclopyr.
The first step cannot be taken until late august/early September due to the timing of moving the chemical to the roots
1. Borrow injector gun from county if one is available. Fill it with *concentrated* glyphosate. Inject each knotweed stem below the first node. You may want some sort of paint spray to mark which ones you’ve sprayed
2. It will die off after step one, but the next year it will come back, much smaller. You can’t inject these stems anymore, so you spray the tops with glyphosate. Again, wait until august/sept.
3. It will all die off again, and then very very few will pop up in the following years. You should continue to spray them individually in august/sept every year until they are completely gone. There will be very little to do as it will be only one or two plants popping up.
There is no other way to remove knotweed. You can’t cut it, you can’t mow it, nothing. Please do not compost it or transport it. You have to burn it once the stems have dried. It is ruthless because of its underground rhizome growth which can extend up to 10 ft below the soil surface. Don’t listen to anyone that tells you chemicals aren’t needed because you will waste time and likely spread it in the process of trying other methods. Remember that even a tiny piece can start a new plant.
[here is what the injector gun might look like](https://snohomishcountywa.gov/2012/Knotweed-Stem-Injector)