Bracket structures in knot theories: from Kauffman bracket to colourings brackets and unsolved problems

Kauffman bracket is one of the most well-known and fundamental invariants of classical knots. In general, the idea of developing a «bracket» invariant from certain smoothings of crossings is very fruitful and it was successfully implemented in many settings.
An interesting example of bracket invariants is the parity bracket defined by V.O. Manturov. The parity bracket was used by its author to prove that the theory of free knots is non-trivial. Moreover, this bracket was the first working example of picture-valued invariants: the invariant value of the bracket was not a number or a polynomial, but a “picture” — an actual graph (a concrete knot diagram, not a class of diagrams).
Later, inspired by the work [2] of S. Nelson et. al., Ilyutko and Manturov [1] defined the parity-biquandle bracket, which dominated both Manturov parity bracket and Nelson biquandle bracket. Later on, the author together with Manturov and Nikonov developed a colourings bracket which in a way may be regarded as an «ultimate generalisation» of the former ones.
These results lead to a natural desire to shift bracket constructions by a dimension: to define them in the case of 2-dimensional knots.
In the talk I will give an overview of these concepts, and present a framework of unsolved problems, the most central of which is: we have the machinery of smoothings for 2-dimensional knots; how do we define bracket structures for them?
If time permits, the smoothing techniques in the setting of 2-knots will be discussed in detail.
The talk is based on a joint work with I.M. Nikonov and V.O. Manturov. The work was partially supported by RFBR grants 20-51-53022, 19-51-51004, and 19-01-00775.