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 kR⁶ᵏ ℝ 6 k

We derive combinatorial formulas for invariants of smooth embeddings of left parenthesis 2 script l minus 1 right parenthesis (2-1) ( 2 ℓ − 1 ) -spheres into double struck upper R 3 script l R³ ℝ 3 ℓ for script l greater than or equals 2 ≥ 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 S⁴ᵏ⁻¹ R⁶ᵏ 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 Rn+q ℝ