Like the above, if you check your bag, security should not be an issue. There may be a few other concerns:
1. Each airline has their own additional baggage charges. (except South West) If it is a normal looking hard suitcase, you may pay
$15-20 dollars U.S. each way.* If it is a soft bag(s), and they know it is a bike, they may charge you as high as
$175.00 U.S. for each bag, each way. Check with your airline carrier to obtain the correct charges.
2. Any bicycle accessories, that you value, I would take on the airplane, in your carry on bag. (minus any tools)
I would check with your airline or the TSA, online, for a list of prohibited items.
3. In lieu of the Christmas Day incident, people are nervous around airports. If I were you, set your bike up outside of the airport.
4. Riding from JFK to Manhattan can be a challenge. I believe you can ride over the Brooklyn Bridge but I would research, very carefully, a safe route, into the city. I hope this helps and you have a wonderful trip:)
*If you go online, to the airline's website, you could save a few dollars, by pre-checking in your bag(s)