This is a tricky grey area.
In general I agree with @Sard. But I also feel compelled to mention this:
You are probably more accurate saying that "cold-carry" is not illegal, that in itself doesn't make it legal. And carrying a piece of paper with the law printed on it is not necessarily going to impress a law enforcement officer - if they cannot book you on carrying a weapon they may just find something else instead.
You may also find that carrying a weapon isn't illegal*, but using that weapon is. Remember that in general being a martial artist doesn't infer special rights on your or make you any different to a normal member of the public. If you carry a weapon you are far more likely to get into a situation that you should have avoided.
So, do you really need to "cold-carry" that weapon? Or does it just make you feel better?
*For example it may be illegal unless you have a legal purpose for carrying it, i.e. you are transporting the weapon to martial arts training.
In this country I can go and train in the local park with a live katana, but if a police officer found me just carrying one in my car for no real reason then he has grounds to book me ("self defense" is not an adequate reason).