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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.14668 |
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| _version_ | 1866912716686360576 |
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| author | Gauniyal, Neeti Turchin, Victor |
| author_facet | Gauniyal, Neeti Turchin, Victor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14668 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Goussarov-Polyak-Viro type formulas for $(4k-1)$-dimensional knots and links in $\mathbb{R}^{6k}$ Gauniyal, Neeti Turchin, Victor Geometric Topology Algebraic Topology Quantum Algebra 57R40, 57R42, 58D10 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. |
| title | Goussarov-Polyak-Viro type formulas for $(4k-1)$-dimensional knots and links in $\mathbb{R}^{6k}$ |
| topic | Geometric Topology Algebraic Topology Quantum Algebra 57R40, 57R42, 58D10 |
| url | https://arxiv.org/abs/2511.14668 |