knot theory - Higher order quandle

For any codimension 2 embedding that is locally flat, there is a notion of the fundamental rack (not every element is idempotent). In Euclidean space, there is a fundamental quandle. It is defined as in the classical case: homotopy classes of paths that start on a tubular neighborhood and that end at the base point. The top of the homotopy is required to remain at the base point, but the bottom stays on the tubular neighborhood. More specifically, two paths $a_0$ and $a_1$ with $a_j(0) in partial (N(K))$ and $a_j(1)= {mbox{ pt.}}$ are equivalent if $exists: A:[0,1]times [0,1] rightarrow {bf R}^{n+2}$ such that $A(j,t)=a_j(t)$ for $j=0,1$, while $A(0,s)in N(K)$ and $A(1,s)={mbox{pt.}}$

The quandle operation is to follow $a$ to the basepoint, go down $b$ around the meridian at $b$ (in an oriented fashion), and return to the base point at $b$. See this manuscript (in particular the illustration therein) to see what is going on.

In additional to the fundamental quandle, there are quandle cocycle invariants that detect a lot of things about classical knots and knotted surfaces. For classical knots, the 2-cocycle invariant is related to a choice of longitude. See
M. Eisermann. Homological characterization of the unknot. J. Pure Appl. Algebra, 177(2):131–157, 2003 for this result.

Fenn and Rourke showed that the 3-cocycle invariant for classical knots could detect the chirality of the trefoil.

The 3-cocycle invariant for knotted surfaces (and its generalizations) are known to give strong results about non-invertibility, bounds on triple point numbers, and in the case of symmetric quandle homology, they give bounds on the triple point numbers of non-orientable surfaces.

In higher dimensions, one can prove that higher cocycle invariants exist --- there are generalizations of the Roseman moves that are given via multi-germs (due to Mond and another author), and the quandle $n$-cocycle condition corresponds to the $n$-simplex move in this context. There is a lot of work that can be done in this regard. I think most of it is straight-forward.