Is there a non-trivial knot with trivial Homfly polynomial?

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/