Depends on where you live. Knife laws in Ohio, unlike gun laws, can vary from town to town. What's legal in Akron may not be in Cleveland or Columbus. Just about everything is OK to own/possess but carrying them can be a gray area. You cannot carry any "deadly weapon" CONCEALED in Ohio other than a handgun with CCW permit. So short blades that you can reasonably say are used as a "tool" are ok. Basically, Ohio laws are such that folding and fixed blades are OK to carry concealed if under a certain length (length varies based on municipality) provided you're not carrying them as a weapon. Switchblades, balisongs, and gravity knives are not allowed for concealed carry as they are principally classified to be "deadly weapons". Knives which are not suitable for CONCEALED carry can be OPENLY carried (not concealed) for "agricultural, hunting or utiliy" purposes. Spring assisted opening knives (not to be confused with automatic knives) are not considered switchblades and legal to carry concealed provided all the other conditions are satisfied.
Short knives that are not weapons can be carried concealed as "tools". Legal length varies by local laws - typically 2.5 inches.
Long knives, balisongs, gravity knives, and switchblades are deadly weapons and can only be open carried for utility purposes.
Spring assist knives without a switch are not considered switchblades.
Basically, what it really comes down to is the law is so vauge that you could find yourself in court for carrying just about any knife in any fashion based on a cop's perception of why/how you're carrying it - Ohio laws are all about INTENT. Don't sweat it. If you choose to carry a knife just keep your nose clean and don't give cops a reason to search you. Don't make trouble and The Man won't make trouble for you. I've carried knives that are technically not allowed (balis, switchblades, big folders, etc...) for years and never had an issue - then again I'm a responsible/respectable looking adult and keep out of trouble/suspicion. Plus, I'm permitted to carry a handgun so I doubt a cop with even an ounce of common sense would care that a knife I have is not allowed when I also have a handgun.