An n-component link is said to be ribbon if it bounds a ribbon surface consisting of n discs. (a ribbon surface is an immersed surface with only ribbon singularities, see http://en.wikipedia.org/wiki/Ribbon_knot).
Let $L=L_1\cup,...,\cup L_n$ be an n-component link (n>1). The following conditions are necessary for L to be ribbon:
- for each pair $(L_i,L_j)$ we have $lk(L_i,L_j)=0$ (lk=linking number)
- each $L_i$ is itself a ribbon knot
- Let V(L) be the Jones Polynomial of L and $V(O^n)$ be the Jones polynomial of the trivial n-component link then $V(O^n)$ is a factor of V(L) i.e. $V(O^n)|V(L)$ (Eisermann arxiv.org/abs/0802.2287)
My questions are:
Is there any example of a link which is not ribbon satisfying the previous conditions?
Are there any other necessary conditions?