We have no topologists on our faculty, and from time to time I get to teach our topology course. I know that there are examples of inequivalent knots with the same Homfly polynomial, and I know that there are non-trivial knots with trivial Alexander polynomial, but I don't know whether the question has been settled as to whether there is a non-trivial knot (or link?) with a trivial Homfly polynomial. I'd like to give my students up-to-date information on this, and being outside the area I don't know quite where to look.
[Math] non-trivial knot with trivial Homfly polynomial
knot-theory
Best Answer
All I know is that in their 2003 paper Eliahou, Kauffman and Thistlethwaite write that they did not find any links with trivial HOMFLY-PT. Although they do find links with both trivial Jones and Alexander.
http://www.math.uic.edu/~kauffman/ekt.pdf
My guess would be there exist links with trivial HOMFLY-PT but no such knots.
Although as Qiaochu mentions it is still open whether there are knots with trivial Jones, Joergen Andersen claims on his website that there are no knots with trivial Colored Jones. (a.k.a Jones of all the cables of the knot)
http://home.imf.au.dk/andersen/