Consider a finite $n$-element classical (real) link and the resulting link structure obtained by cutting each of the component elements (knots). Let us represent the resulting structures in a tableau, where each term gives the cut element and a list of the resulting links made from the remaining component elements.

For instance, in a $3$-element Borromean ring (in which cutting any component leads to separation of *every* element, $6_2^3$ in Rolfsen's table of links) the tableau would be:

- $l_1 \to \{ l_2, l_3 \}$
- $l_2 \to \{ l_1, l_3 \}$
- $l_3 \to \{ l_1, l_2 \}$

A fully interlinked link ($6_3^3$ in Rolfsen's table of links) would have the tableau:

- $l_1 \to \{ l_2 l_3 \}$
- $l_2 \to \{ l_1 l_3 \}$
- $l_3 \to \{ l_1 l_3 \}$

A simple $3$-element linear chain (link) would have the tableau:

- $l_1 \to \{ l_2 l_3 \}$
- $l_2 \to \{ l_1, l_3 \}$
- $l_3 \to \{ l_1 l_2 \}$

where of course here multiplication denotes linked elements.

Call such a tableau "feasible" if there exists a corresponding $n$-element link.

**Questions**

- Is there an algebraic method to determine if an arbitrary tableau is feasible? For instance can one algebraically show whether the following tableau is feasible?

• $l_1 \to \{ l_2 l_3 \}$

• $l_2 \to \{ l_1 l_3 \}$

• $l_3 \to \{ l_1, l_2 \}$

- Assuming a tableau is feasible, is there an algorithm to construct a corresponding link? (Of course the corresponding link need not be unique, up to ambient isotopy.)
- What is the number of feasible tableaus (up to permutation symmetry of the labels of the component links) for a given $n$?