If you signed an actual written contract requiring you to give 4 weeks notice, then if you fail to do so and if the fact that you provided insufficient notice somehow provably costs your employer money (e.g. to hire a temp or freelancer at a premium to do your job for the time you should have given notice), the employer could sue you for those costs or losses based on your "breach of contract." But that's only if you had a written contract. In the absence of a contract, their policy is not enforceable and you could give 2 weeks notice, or even no notice at all.
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