The Blade Dodge Takedown is a Takedown available from Batman: Arkham City onwards. When faced with an enemy carrying a blade or sharp object, the player can turn the tables on them by dodging each attack and finishing them with a devastating blow that immediately knocks the opponeny unconscious.
The Blade Dodge Takedown can be difficult to master. Most players probably just tap the counter button, which makes it even harder to pull off consistently, while the key to doing it is to hold the button and release it after each of the enemy's swipes. Try to learn the simpler Blade Dodge first.
The Blade Dodge Takedown adds 3 multipliers (only 2 if the character hasn't entered Freeflow mode, i.e. at the very first attack as shown in the gif above, just like a glide kick, a strike while running, or a trip slide does) to the combo meter and rewards 100 points for each multiplier in the current combo, the highest amount of points for an attack due to its difficulty and uncontrollable nature: like counters, players can't actively perform it, only can do so in response to an enemy attack. As the move gives more points than any other attack (except the Titan Takedown and Environment Takedowns), it's useful to leave knives undestroyed and learning to perform the move to earn some extra points during challenges. The Blade Dodge Takedown does not count as a combat variation, but only as a form of the counter.
The Blade Dodge Takedown does not destroy an enemy's blade or bottle. When the enemy is knocked unconscious, their weapon merely fall to the floor, where another enemy could pick it up and use it. They are vulnerable to the Special Combo Disarm and Destroy, if you wish to destroy the enemy's weapon for the duration of the battle.
The Takedown requires better timing in Arkham Knight than in City or Origins. A Blade Dodge Takedown can be used against a brute with blades, although it results in a mere Instant Takedown (only 50 points multiplied by the current combo in combat challenge maps); an armored brute requires two takedowns to be completely knocked out.
The Takedown can be used on any type of enemies with broken bottles or sharp blades (thugs, ninjas/combat experts, brutes, except Sickle), but also on Copperhead in Batman: Arkham Origins as she attacks with her sharp fingernails.