As it relates to 分かる and 知る, you can think of it like 分かる has an implication of your personal knowledge, the things that relate to you, the things a person can decide on, etc... Only you can "know" what you do tomorrow, or decide on it, or reflect on it.
Whereas 知る is for the things that are facts independent of you, like the atomic weight of cesium, what the airspeed velocity of a sparrow is, how Korean and Japanese chopsticks differ, etc...
As a result, there can be an implication if "I don't care" interwoven into the use of 知らない in some situations.
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