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

TL;DR

This work extends the GPV framework to higher-dimensional knot theory by deriving combinatorial, Gauss-diagram–style formulas for invariants of embeddings $S^{2\ell-1}\hookrightarrow \mathbb{R}^{3\ell}$ and a concrete formula for the Haefliger invariant $\mathcal{H}:\pi_{0}Emb(S^{4k-1},\mathbb{R}^{6k})\to\mathbb{Z}$ under generic projections. The core method projects high-dimensional knots to a hyperplane, analyzes the preimages of double points, and encodes crossing data via higher-dimensional linking numbers and the Ekholm orientation, yielding explicit invariants for 2- and 3-component links and a full invariant for Haefliger knots. The paper also connects these invariants to the Smale invariant of immersions through $E(f)$ and $\Omega_{S^{2k}}(f)$, proves Borromean-type generators' behavior, and demonstrates a configuration-space viewpoint with a homotopy-retract result for spaces of braids, addressing a conjecture in the field. Overall, the results broaden the finite-type invariant philosophy to high dimensions and illuminate the interactions between projection data, linkings, and high-dimensional knot isotopy classifications."

Abstract

We produce combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. 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 $\underset{k}{\sqcup}\mathbb{R}^n\hookrightarrow\mathbb{R}^{n+q}$ for $q\geq 3$, thus proving a conjecture of Komendarczyk, Koytcheff and Volić.

Figures (28)

• Figure 1.1: Gauss diagram of $3_1$ knot
• Figure 1.2: Another description of the Gauss diagram of $3_1$ knot
• Figure 2.1: Possible components of $\overline{L(f)}$ when $\ell=2$
• Figure 2.2: Corresponding $f(\overline{L(f)})$ for $\ell=2$
• Figure 2.3: First Reidemeister move for $M^3\hookrightarrow \mathbb{R}^6$

Theorems & Definitions (30)

• Proposition 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• Remark 3.5
• Example 3.6
• Example 3.7