I think I can answer this one.
I actually control this noxious invasive plant as part of my work and got my pesticide applicator's license just for the purpose of getting rid of this stuff. We have it ALL over the banks of the West Branch of the Susquehanna river and it chokes out a ton of very important riverside habitat.
My organization's recommendations call for cutting the tall canes as near to June first as you can, then waiting 6 weeks and hitting the resprouts with some sort of glyphosate herbicide (Roundup is the one most homeowners can buy). Apply the herbicide to the the resprouts and make sure to 'spray till wet' meaning, get all of the leaf surface glistening with the chemical. Make sure you follow the label instructions on when and how to apply. This procedure has had the best results in our trials, but don't expect to get everything the first year. You will likely have to repeat this procedure for a couple of years before you can completely eradicate this pesky weed. The Japanese knotweed has an ENORMOUS root system, once established, and you need to deplete the stored energy in that root system in order to completely kill it. And once you have gotten rid of it, keep alert for reintroductions. All it takes is one small piece of this plant (it tends to reproduce vegetatively rather than by seeds) to start the whole process over again.
The good news is, we are on the verge of developing some integrated pest management strategies to control this weed. Though there are still some tests to do, we might have a way to get some biological control on it soon. Stay tuned to your local Ag extension or Forestry office for further info.
FYI -
http://www.dcnr.state.pa.us/cs/groups/public/documents/document/dcnr_010254.pdf