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

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 \(\m...

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Bibliographic Details
Published inarXiv.org
Main Authors Nozaki, Yuta, Sato, Masatoshi, Suzuki, Masaaki
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.01.2022
Subjects
Colon
Group theory
Homology
Kernels
Mathematics - Geometric Topology
Mathematics - Quantum Algebra
Quotients
Surgery
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Summary: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 \(n \geq 2\).
ISSN:2331-8422
DOI:10.48550/arxiv.2103.07086