N-tangle Kanenobu knots with the same Jones polynomials

Abstract

It is still an open question if there exists a non-trivial knot whose Jones polynomial is trivial. One way of attacking this problem is to develop a mutation on knots which keeps the Jones polynomial unchanged yet alters the knot itself. Using such a mutation; Eliahou, Kauffmann and Thistlethwaite answered, affirmatively, the analogous question for links with two or more components. In a paper of Kanenobu, two types of families of knots are presented: a 2- parameter family and an n-parameter family for n ≥ 3. Watson introduced braid actions for a generalized mutation and used it on the (general) 2-tangle version of the former family. We will use it on the n-tangle version of the latter. This will give rise to a new method of generating pairs of prime knots which share the same Jones polynomial but are distinguishable by their HOMFLY polynomials.