My sister is afraid to let me have a link on her Facebook page (I don't use Facebook) to my page at the artofwhere website where I have products adorned with my artwork for sale. Am I correct that her page cannot be hacked by way of that link to my page?
You are correct. Under normal conditions, simply putting a link on a page is insufficient to hack that page. It may be a problem to put a malicious link on an insecure website, if said link is specially crafted to exploit a bug like XSS, but that is not an issue if you simply put your link on her page.