Matthew,
I've just joined the site, but I was reading that you had no replies to your spreading knotty weed problem, and thought you might try this...
Unless you have strong opposition to chemicals, I would shroud the area surrounding several clumps of the weed with newspaper or cardboard or the like, and spray the green areas with a systemic weed killer like Roundup. Allow this to dry before removing the surrounding protection, then your problem should take care of itself in a few weeks, and the herbicide spreads through the root system.
Hope this helps, if you hadn't already found the solution...
Kathie