connection, barrier, gateway, involve, concerning
関 centers on a point of connection or passage that can also act as a barrier: a gateway that links or separates, involvement that ties things together, and matters concerning something.
relation; relationship; connection
relation; connection; relevance
regarding ...; concerning ...; about ...; on ...; with respect to ...; in regard to ...
engine
to be involved (in); to get involved (in); to concern oneself (with); to take part (in); to interfere (in)
Kansai (region comprising Kyoto, Osaka, Kobe and surrounding prefectures)
Kantō (region comprising Tokyo and surrounding prefectures)
concern; interest
barrier; gate
participation; taking part in; participating in; being concerned in
entrance; front door; entryway; entranceway; entry hall; vestibule; porch; foyer; mud room
joint
customs (duties); duty; tariff
correlation; interrelation
customs; customs house
barrier; obstacle; difficulty; hurdle; deadlock
ōzeki (second highest rank in sumo)
to concern; to be related
barrier; checkpoint; gateway
customs clearance
(one's) concern; having (something) to do with
ranking wrestler in the makuuchi (senior-grade) or juryo (junior-grade) divisions
matter of concern; matter of interest
wrestler of the third highest rank
関 is the modern simplified form of 關. The older form combines 門 (gate) with a phonetic component that also suggests a barrier or bolt. The modern 关 inside is a simplification that retains the gate imagery.
The gate 門 encloses a simplified barrier 关. Picture a guarded gateway that controls passage: it connects two sides but also blocks entry, just as involvement connects people but can also create barriers.
For カン, imagine a can of bolts and bars used to secure a gate: can -> カン, and the gate is the barrier that connects or separates.