Martin has 28 career tackles with a half-sack and a forced fumble in 21 NFL games, all with Kansas City. He has also spent time with Tampa Bay.
He replaces Quinton Coples, the Jets' 2012 first-round draft pick who was waived on Monday.
Martin was signed by the Chiefs as an undrafted free agent in 2013 out of Columbia University, where he was an All-Ivy League first-team selection. He spent two seasons in Kansas City before being signed by Tampa Bay in September. He was waived/injured by the Buccaneers last month and signed to the Colts' practice squad on Oct. 31.
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