On the kernel of the surgery map restricted to the 1-loop part

Yuta Nozaki; Masatoshi Sato; Masaaki Suzuki

doi:10.1112/topo.12233

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arXiv · 2103.07086

On the kernel of the surgery map restricted to the 1-loop part

Yuta Nozaki·Masatoshi Sato·Masaaki Suzuki

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Abstract

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map $\mathfrak{s} \colon \mathcal{A}_n^c \to Y_n\mathcal{IC}_{g,1}/Y_{n+1}$ is surjective for $n \geq 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of $\mathfrak{s}$ restricted to the 1-loop part after taking a certain quotient of the target. Also, we introduce refined versions of the AS and STU relations among claspers and study the abelian group $Y_n\mathcal{IC}_{g,1}/Y_{n+2}$ for $n \geq 2$.

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BibTeX RIS

Yuta Nozaki, Masatoshi Sato, Masaaki Suzuki. 2022-01-05. On the kernel of the surgery map restricted to the 1-loop part. https://doi.org/10.1112/topo.12233

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