A common way to produce the ASL sign for "INVESTIGATE" is with a double movement.
When produced with a single movement the meaning changes to "CHECK"





CHECK: The idea behind this sign is to draw a "check" mark on your hand. The actual movement however is very quick and relatively straight. The tip of the right index finger quickly moves down and forward, glancing off the upturned left palm.
.


Note: if you mean "check" as in
"Write a check," -- see "CHECK (1)"
INVESTIGATE:
This version uses more of a digging movement. Think of an archeologist
digging into the dirt to investigate a fossil.






You can initialize this
sign to specifically make it mean "research."
RESEARCH:






Notes:
See: CHECK
See: PROOF
See: STUDY
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