A contract signed under duress is not enforceable.
If the Company wants to collect show them what was originally offered to you versus the revised contract they actually put in front of you after you had already quit your job, turned down other offers, relocated, etc, etc. Tell them you signed their revised contract under duress and that not only is it not enforceable but you are entitled to other damages as well - perhaps multiples of what they are asking. Tell them you would be reluctant but forced to pursue these options with your attorney if they wish to proceed.
In your correspondence, copy your hiring manager, HR and the Company's legal department and ask for a response in how they would like to proceed.