Erez, be sure you're using a quality blade with a tooth configuration appropriate for the job and make sure any blade guides are adjusted properly...your cuts should be just fine for that, staying just away from the line and then cleaning it up with files. You may want to rig up a good fixture to securely hold and support the workpiece. I'll assume you're cutting the slots prior to shaping the back of the neck which will make securing things a lot easier! You want that workpiece secured down tight for such a delicate cut in what is often expensive material.
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The most expensive tool is the one you buy "cheaply" and often...