I have actually fixed Travertine for the same thing a few times.
one was a chiseled edge and we used unsanded matching wall grout,
let it pretty much dry for a while and then used a razor and slide it across
the top and then buffed it out, looked like a vein when we were done.
the other was honed Travertine and we used and epoxy called AKEMI
to color it and then polish it.