Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for knotted surfaces I mean some relations between a projection of a knotted surface on $\mathbb{R}^3$ and various resolutions of its singular loci, analogous to the usual skein relations for knots.

## 2 Answers

It is known that a knotted surface can be presented by a marked graph diagram, which is just a knot diagram while some crossing points are equipped with markers. On the other hand, two marked graph diagrams represent the same knotted surface if and only if they are related by finitely many Yoshikawa moves. Then one can apply the Kauffman skein relations (with some modification) on these marked graph diagrams to define knotted surface invariants. Some concrete examples can be found in “Lee, Sang Youl. Towards invariants of surfaces in 4-space via classical link invariants. Trans. Amer. Math. Soc. 361 (2009), no. 1, 237–265.” and “Lee, Sang Youl. Invariants of surface links in ℝ4 via skein relation. J. Knot Theory Ramifications 17 (2008), no. 4, 439–469.”

The answer might depend a bit on exactly what you want; perhaps giving a precise formulation is the hard part!

There was an important first step in this direction for the Alexander polynomial, by Cole Giller: Towards a classical knot theory for surfaces in $\mathbb{R}^4$. Illinois J. Math. 26 (1982), no. 4, 591–631. There are a lot of references to that paper in the subsequent literature.

You should also look at works of Carter, Kamada, Saito, Ogasa in this direction.