In the divorce/family law field a notarized letter is not worth the, well, paper it is written on. New York has very strict rules about documents. If he has no problem with the move, then you should get an agreement, signed and notarized with the same formality as with a deed for real property.
While the notarized letter may help you in an application for permission to move, it is not legally binding. Dad could object and then you have to get Court permission BEFORE you move or you could be in real hot water and possibly lose custody.
There are differing opinions on this point. I support the idea that a written agreement permitting a relocation is sufficient proof of permission of a relocation. The agreement should be notarized and authenticated.
Other day a petition is needed in family court and the stipulation be converted into an order.
Either way, be sure you gave him adequate substitute parenting time along with holidays and vacations.
If he has truly no problem with you and the child moving to CT, yes, he can sign a consent and have it notarized.
If he is interested in visitation, you may need to deal with some workarounds for the fact that frequent visitation may be more difficult, for instance, allowing longer periods during school vacations, or some arrangement for the custodial parent to pick up more of the transportation costs are often concessions (especially if your divorce decree or prior Family Court orders require the Non-custodial parent to pay child support).
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I agree with Attorney Port. Have it drawn up correctly with the proper full acknowledgment.
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As my colleagues have written, there is no simple answer. A signed notarized letter can help, but it is not dispositive. If you move and three months later the father changes his mind and files, the notarized letter and the fact you have taken substantial steps may play a role. Court permission is usually needed to relocate. The primary case in NY on relocation is TROPEA. You are best advised to check any stipulation of settlement for relocation language and obtain the advise of counsel before taking any steps.