It is well known that $\operatorname{Wh}(\operatorname{Bor})$ link (That is, untwisted Whitehead double of Borromean rings with positive clasps, say) is very interesting.
Are there any easy way to compute $\pi_1(S^3-\operatorname{Wh}(\operatorname{Bor}))$ from the group presentation of $\pi_1(S^3-\operatorname{Bor})$ which is relatively simple?
For example, I do know that $\pi_1(S^3-\operatorname{Bor})$ is isomorphic to the group $\langle a,b,c~|~a=[c,b^{-1}]b[c,b^{-1}]^{-1}, b=[a,c^{-1}]b[a,c^{-1}]^{-1}\rangle$ which is obtained from the 3-bridge notation of Borromean rings in A. Kawauchi's book, A survey of knot theory, Example 6.2.8.