Tilling wont spread this pest, it will probably help you a little though. Its not the seeds that you have to worry about, its the roots. This bad boy can sprout from as small a piece of root as 1 inch, and can live underground covered up by plastic for up to 5 years. The knot weed will just come up arround the plastic.

However, It doesnt like lime,it doesnt like to be mown, and it has a hard time comming up through a healthy bed of grass... ( it will try though, if it has no other option).

My recomendation would be to till up your piece of ground, and remove as many of the roots as you possibly can. Then mix large qty's of lime into the soil, and cover with heavy plastic, and build your raised beds. Lime the "infected area" arround your beds, add grass seed, and weed out any emerging shoots with extreem predjiduce! When your grass starts growing, mow it religiously!

Ive successfully gotten rid of it from my yard using this method.