We replaced our countertops in our kitchen and did not make the hole for the dishwasher hose that goes up to the top of the sink. What is this hose for and can we connect this direct to the drain?

Not an expert here so wait for more advice. We've got some top notch plumbing people.

I think you are talking about an air gap... a dw line that ran from the dishwasher toward the countertop, connecting to a y, then another line from that to the drain or disposal unit. It's a safety feature that prevents water from siphoning from the dishwasher. Not sure if its code.

You might want to post where you live, in case our experts need that to know whether you can get away without it. Also what's the make of the dishwasher. I've heard some makers are installing built-in safety loops to help prevent siphoning, but I don't know enough about that.

I have a dishwasher installed without one and it works fine. Its just a safety feature, and, as I said, might be required by code, or not.

Do the installation instructions for the dishwasher tell you to install to the disposal with a loop... meaning you run the drain waste line up to the highest point under the counter and then loop without kinking to the disposal? Similar idea.

Time for the experts to chime in.

It is an air gap. Just loop the drain hose to the underside of the counter and secure it then back down to where it connects to the drain and it will work fine.