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.
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asked May 24, 2010 at 6:27
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1$\begingroup$ Wikipedia says it's open whether there even exists a nontrivial knot with trivial Jones polynomial, although of course it's unclear how up to date that information is. $\endgroup$– Qiaochu YuanMay 24, 2010 at 6:35
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1$\begingroup$ That's true to the best of my knowledge. On the other hand, it's now known no knot has trivial HOMFLY homology by work of Mrowka and Kronheimer. $\endgroup$– Ben Webster ♦May 24, 2010 at 12:58
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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)
answered May 24, 2010 at 9:58
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1$\begingroup$ You might be interested in these results of Matthew Hedden and Liam Watson about Khovanov homology and its colored variants detecting the unknot: front.math.ucdavis.edu/0805.4423 and front.math.ucdavis.edu/0805.4418 $\endgroup$ May 24, 2010 at 14:48
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1$\begingroup$ My thanks to all who responded to my question. I hope my acceptance of an answer will not stop anyone who has more to say from posting. $\endgroup$ May 25, 2010 at 0:20
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3$\begingroup$ Kronheimer & Mrowka have announced that Khovanov homology detects the unknot: msri.org/communications/vmath/VMathVideos/VideoInfo/4633/… $\endgroup$– Ian AgolMay 25, 2010 at 1:53
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2$\begingroup$ The link to Kronheimer & Mrowka's preprint is here arxiv.org/abs/1005.4346 $\endgroup$– j.c.May 25, 2010 at 20:11