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Goussarov-Polyak-Viro type formulas for
$(4k-1)$-dimensional knots and links in 
$\mathbb{R}^{6k}$
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 69 / Issue 4 / November 2026
- Published online by Cambridge University Press:
- 25 September 2026, pp. 1466-1505
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- Article
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We derive combinatorial formulas for invariants of smooth embeddings of
$(2\ell-1)$-spheres into 
$\mathbb{R}^{3\ell}$ for 
$\ell\geq 2$. In particular, 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, who expressed finite-type invariants of classical knots in terms of Gauss diagrams. Analogously, we project higher-dimensional knots and links onto a hyperplane and study the preimages of the loci of double and singular points in the embedded spheres. As an auxiliary result, we show that the space of 
$n$-dimensional braids with 
$m$ strands in 
$\mathbb{R}^{n+q}$ is a homotopy retract of the space of string links 
$\underset{m}{\sqcup}\mathbb{R}^n\hookrightarrow\mathbb{R}^{n+q}$ for 
$q\geq 3$, thereby providing a short proof to a conjecture of Komendarczyk, Koytcheff, and Volić.
