TITLE: Links with same Jones polynomial
QUESTION [5 upvotes]: 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?

REPLY [5 votes]: Liam Watson generalized a construction of Kanenobu to produce infinitely many pairs of knots with the same Jones polynomial (and Khovanov homology) but distinct HOMFLYPT polynomials, so they are not mutants.  See the references below.
MR2287438 Watson, Liam.  Any tangle extends to non-mutant knots with the same Jones polynomial. J. Knot Theory Ramifications 15 (2006), no. 9, 1153–1162. 
MR2350287 Watson, Liam.  Knots with identical Khovanov homology. Algebr. Geom. Topol. 7 (2007), 1389–1407.