First of all, rhetorical questions: What is a "throwing knife"? During the process of knife manufacture, starting with shape cutting and ending with sharpening, at what point do we have a knife? The commonly accepted definition of a "knife" is something with a sharp cutting edge or blade. By this definition, a "throwing knife" with an unsharpened edge is not a knife. It is a pointy piece of metal used for recreation. I'm not a lawyer and my intention isn't to argue these points - just something to think about.
Anyway, most laws look at three things:
(1) Is the weapon specifically banned? The list usually includes automatic knives, stars, blackjacks, billy clubs, brass knuckles, etc.
(2) Is there criminal intent? This is somewhat circular because some laws assume criminal intent if point (1) is met, much like drug laws.
(3) Does the weapon have a generally accepted lawful use, or is there some other acceptable excuse?
Carrying weapons in public is very different than using them on your property.
Read this
http://knife-expert.com/ny.txt (change "ny" in the url to see another state). I skimmed over it, but it looks like NY is pretty tolerant of throwing knives.