I figure, if it's going to be here hogging up the landscape, we might as well put it to work.
This is an experiment, I don't know if it's going to really work yet, but I think it has potential.
First progress photos added: 5/24/13
Okay, so a lot of people are asking: What is the point of this? Well, it gets plants you want to be growing (instead of the knotweed) up into the sunlight. The problem is that the knotweed blocks out the light from the ground so no other plants can get started. This lets you grow other plants on top of it, sort of defeating its evolutionary advantage. You're basically using the knotweed to help other plants you prefer to grow get advantageous access to sunlight. I hope that helps explain it better for those who don't get the point.
Another question people are asking is: Why don't you just kill the plant? Well, it's also on the neighbors yard primarily and I think they like it for the privacy screening it provides. It's not exactly a problem. Believe me, the real problem is blackberries. But anyway. I only knew of its noxious weed status after a visit from the local arborist who kind of freaked out about it. But If it gets to be a problem, I would probably choose the method suggested by HSLINKS which is to cover it completely in three layers of carpets.
But the best news is: this plant is very much edible! As pointed out by user Hoopajoo:
See also: http://www.wildmanstevebrill.com/Plants.Folder/Knotweed.html
Here's a recipe: http://www.richmondlandtrust.org/docs/garlic.pdf
And another: http://eastoneccentric.blogspot.com/2013/05/if-you-cant-beat-it-eat-it-japanese.html
Okay, I tried it raw as described in the video and while I would not call it delicious, it is decidedly lemony and vegetably and a case could be made for sour-apple. It wasn't entirely unpleasant and I did not get sick or die, so I'm glad to know about one more wild edible in case I'm ever out in the woods and hungry. I think though that the key may be to cook it. I'm going to try that soon. Interestingly, it is also sold in supplement form as a source of resveratrol.
Be careful of course, make sure you know you have the right plant. There are plants out there with mottled green/red stems which are poisonous such as hemlock