I promised to show some photos trying to illustrate the usage of my cut knurling tool. I don't have any dimensions as I just dig up some good looking pieces of steel and started to machine. I think anyone can figure their own version with possible improvements and I am happy to help if there something mystery.

These are the main parts:

Knurling tool set to toolpost, mounting shaft angled approx 5-degrees in the tool post to create relief angle for the cutting edge:
(In this picture the Knurling tool set to cut the second cut of the diamond pattern, lathe run in reverse)

Centering the knurling tool to workpiece with ruler method:

Position for cutting lathe spinning in normal direction:

Rest of the photos are here:

You really need to flush the tool with coolant or use constant air blast to blow of all the chips so that they don't mangle between the tool and workpiece.