Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing *I* would try is to compute the (formal - you would have something\*split-horizontal+something\*split-vertical) Jones polynomial of (4-)tangles and look *there* for two tangles with same Jones polynomial. Any instance would then generate an infinite example family. The snag would be, of course, that I'd first need a tangle table - my own goes to meagre 6 crossings. Aaaand the computation can't be automated that good. Still - any takers?



Links with same Jones polynomial
Probably the best way to produce infinite families of links with same Jones polynomial is by Conway mutation, this operation does not alter the HOMFLY polynomial either. A good example
of this is given by the family of pretzel links. Take a look at the answer of a question of mine here:


[How to distinguish Pretzel links with the same coefficients?](https://mathoverflow.net/questions/46888/how-to-distinguish-pretzel-links-with-the-same-coefficients)


By using satellites of the Hopf link it is possible to produce an infinite family of links with the same Jones polynomial of the trivial link. This was done by Shalom Eliahou, Louis H. Kauﬀman and Morwen B. Thistlethwaite:


<http://homepages.math.uic.edu/~kauffman/ekt.pdf>


