Goussarov-Polyak-Viro type formulas for left parenthesis 4 k minus 1 right parenthesis$(4k-1)$ ( 4 k − 1 ) -dimensional knots and links in double struck upper R 6 k$\mathbb{R}^{6k}$ ℝ 6 k

Abstract We derive combinatorial formulas for invariants of smooth embeddings of left parenthesis 2 script l minus 1 right parenthesis $(2\ell-1)$ ( 2 ℓ − 1 ) -spheres into double struck upper R 3 script l $\mathbb{R}^{3\ell}$ ℝ 3 ℓ for script l greater than or equals 2 $\ell\geq 2$ ℓ ≥ 2 . In particular, we obtain such a formula for the Haefliger invariant, which classifies smooth knots upper S 4 k minus 1 right arrow with hook double struck upper R 6 k $S^{4k-1}\hookrightarrow \mathbb{R}^{6k}$ S 4 k − 1 ↪ ℝ 6 k 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 $n$ n -dimensional braids with m $m$ m strands in double struck upper R n plus q $\mathbb{R}^{n+q}$ ℝ