If someone with hepatitis b washed their hands in the sink after cutting themselves with no visible blood in the sink would the soap and water wash away traces of blood in the sink base?

The soap and water would wash away Most but maybe not all of the blood. The Hepatitis B virus is MUCH smaller and could still be present even if no blood could be seen in the sink. I would wear waterproof gloves and decontaminate the sink several times before I would use it myself. Make sure the product that you are using says that it will kill hepatitis B viruses.