Brunnian links are well known, where deleting any component allows you to isotope the rest to an unlink. It's common to construct them by taking an $n-1$ component unlink and defining the $n$th component as a chain of commutators in the Wirtinger presentation such as $[x\_1,[x\_2,[x\_3,x\_4]]]]$


Can you construct $n \ge 4$ component links for any $n$ where deleting any two components leaves an unlink, but the link will not be split after deleting any single component?



Link such that deleting any two components leaves an unlink
Yes, this is done in 


*Penney, D.E.*, [**Generalized Brunnian links**](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077378123), Duke Math. J. 36, 31-32 (1969). [ZBL0176.22201](https://zbmath.org/?q=an:0176.22201).


Call a link *$(n,k)$-Brunnian* if it has $n$ components, and every sublink with $m$ components: does not split when $k<m\le n$; completely splits when $1\le m\le k$. Then it is shown in the above paper how to use iterated commutators to build an $(n+1,k)$-Brunnian link from an $(n,k)$-Brunnian link. 


