I'm going to remove your links and just add a ref to the paper I linked above.  The unknotting problem is a very special case of the more general recognition problem which is also algorithmically decideable.  Also the link to the blog is just a bad idea.  The blog post is a confused discussion of the erroneous MathWorld snippet.  I'd spend some time giving a better description of these issues and format the refs, but I'm planning on rewriting the whole article anyway (got started earlier today).   (Talk)