The Knot Book by Colin Adams has had very good reviews. It claims to need nothing more than high school mathematics as a prerequisite (though one undergraduate reviewer on the Amazon site found it hard going). I haven't seen the book myself, but the review on the American Math. Soc.'s site (pdf download at http://www.ams.org/notices/200508/rev-sossinsky.pdf) shows that it starts at a fairly gentle pace from the very beginnings of the subject, and goes more or less as far as you wnat to take it. You'll certainly find ample material there for an extended essay.