That sort of logic is silly. How do you know he doesn't live in a rural area or participate in outdoor activities? Maybe he needs to cut hay bales, open bags, cut strings, shave the hair on his left arm or any of the other myriad things you do with a pocket knife. I carried a pocket knife at 13. Hell, my grandpa gave me my first knife when I was 7 or 8, although it was a Barlow.
To the op, as others have said, have a frank discussion with your mother. Be respectful and listen to what she has to say, address her concerns. Show her the knives you're interested in and explain to her your reasons for needing it. But remember, she is your mother and you are 13. If she decides that your wants are not warranted, that's her call and you'll have to abide by it. And you may have to do a little proving before she'll give you what you want. You said you leave the knife she had a problem with at your dad's house? Does that imply that he is ok with you having knives? If so, maybe include him in the talk with your mom as his word may carry more weight.