Goussarov-Polyak-Viro type formulas for $(4k-1)$-dimensional knots and links in ${\mathbb{R}}^{6k}$

We produce combinatorial formulas for invariants of smooth embeddings of $(2\mathrm{\ell }-1)$-spheres into ${\mathbb{R}}^{3\mathrm{\ell }}$ for $\mathrm{\ell }\ge 2$. Furthermore, we obtain such a formula for the Haefliger invariant, which classifies smooth knots $S^{4k-1}\hookrightarrow {\mathbb{R}}^{6k}$ up to isotopy. Our approach is similar in spirit to the work of Goussarov, Polyak, and Viro expressing finite-type invariants of classical knots in terms of Gauss diagrams. We similarly project higher dimensional knots and links onto a hyperplane and study the preimages of the sets of double and singular points in the embedded spheres. As an auxiliary result, we show that the space of $n$-dimensional braids with $k$ strands in ${\mathbb{R}}^{n+q}$ is a homotopy retract of the space of long links ${\mathbb{R}}^n\hookrightarrow {\mathbb{R}}^{n+q}$ for $q\ge 3$, thus proving a conjecture of Komendarczyk, Koytcheff and Volić.