Knot polynomials are one of the most common tools that allow us to distinguish between two given knots. And when I say 'knots' I mean one-dimensional knots embedded within $S^3$. I would like to know if there are analogous definitions for higher-dimensional knots, that is, embeddings of $S^{n}$ into $S^{n+2}$ for $n > 1$. Any references are welcome.
Here's one thought about it. If I have a knot in $S^{n+2}$ then I can project the knot onto $\mathbb{R}^{n+1}$. Crossings will likely still make sense, and Google tells me that higher-dimensional analogues of Reidemeister moves exist.[1] It thus seems plausible to me that one can construct analogues of bracket polynomials by means of skein relations.
- [1]: D. Roseman, Elementary moves for higher dimensional knots.
The Alexander polynomial makes sense for codimension 2 knots in any dimension, with the same definition. I doubt there is any Jones-like polynomial. Certainly none is known.