Well, i understand my title is a bit confusing. I will state it clearly below with the example.
Actually, URLs with tilde get converted to absolute URL thanks to the methods : ResolveURL and ResolveClientURL
Therefore you should be able to do this :
" style="border-width:0px;">
(this is actually done automatically for you in Web Controls like HyperLink and such)
The big difference with those two methods happens when you use User-controls. In a case, the URL refers to the URL relative to the folder where the user-control is, in the other case, that would be the page containing the user-control.
See also this question : Control.ResolveUrl versus Control.ResolveClientUrl versus VirtualPathUtility.ToAbsolute