Im 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  Id spend some time giving a better description of these issues and format the refs but Im planning on rewriting the whole article anyway got started earlier today