Some new examples of links with the same polynomials

We call a link (knot) L to be strongly Jones (respectively, Homily) undetectable, if there are infinitely many links which are not isotopic to L but share the same Jones (respectively, Homily) polynomial as L. We reconstruct Kanenobu's knot (Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158-162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: 8(8), 8(9), 10(22), 10(35), 10(155), 4(1)#4(1), 5(2)#5(2)! (5(2)! is the mirror image of 5(2)) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homily undetectable.