Dick,
Truly a thing of beauty... nice job!
I know of no cordage that will not soil, when used in this way. But, I do have an idea for you to try that might suit your needs. After you've completed your knot work, try coating it with West System Epoxy, using a #207 hardener, which gives a non-yellowing, clear coating. Do as many coats as necessary to fully waterproof (and dirt-proof) the cordage. You will probably be happier with a synthetic cordage, as the epoxy is likely to make cotton strands stand up, like the first coat of varnish raises the grain of wood.
I would first try it on a sample piece to see if you like the results. You will no longer have nice feel of the original knot work, but it should be a cinch to keep it clean. Hope it works out for you.
Jack