Video of the incident shows the moment the suspect and three other men made contact as they passed on a stairway.
It happened on Sunday at the West 4th Street-Washington Square subway station.
Authorities say the 28-year-old victim threw a beverage can during the course of the argument, and the suspect then took out a large knife and slashed the victim on his left hand.
The attacker got away on a northbound A or C train.
The victim was treated at the scene by EMS.
Anyone with information in regard to this incident is asked to call the NYPD's Crime Stoppers Hotline at 1-800-577-TIPS (8477) or for Spanish, 1-888-57-PISTA (74782).
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