shut, seal
緘 centers on closing something securely: shutting a container or sealing a letter so it cannot be opened casually.
緘 combines 糸 (thread) with 咸, which likely contributes the sound. The idea is sealing or shutting with thread, as in tying a bag or letter closed.
Thread 糸 on the left and 咸 on the right: imagine using thread to sew a bag shut, sealing it completely.
For カン, picture a can being sealed with thread: can -> カン, and the thread keeps the can shut.