From one finite group $G$, we can build the subfactor $(R \subset R \rtimes G)$ which remembers the group.
Question: Can we build a subfactor planar algebra from one knot? which remembers the knot?
I put this picture to illustrate the question and stimulate the intuition, but I don't ask about this particular knot.
My first naive idea is to build a tangle $T$ from the knot by replacing the n crossings by $2$-box places; and next by distributing two $2$-box generators $u$ and $d$ in the places in such a way that we can recover the data of the initial knot. And finally we consider the planar algebra generated by $u$ and $d$, and at least the relation $Z_T(a_1 \otimes \cdots \otimes a_n) = c_K$ with $a_i \in \{u , d\}$ fixed as above and a fixed well-chosen $c_K \in P_{0,+} = \mathbb{C}$.
A priori such a relation is not enough for getting a subfactor planar algebra; moreover, it is not clear to me what additional relations are necessary, and if they exist such that the knot is finally remembered.
The question above is not necessarily about this idea which is just a clue that something should be possible.
Remark: via a knot polynomial and its Galois group, we can get a group subfactor from a knot, but in general, it does not remember the knot.
