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

Abstract

Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map $\mathfrak{s}:{\mathcal{A}}_n^c\to Y_n{\mathcal{I}\mathcal{C}}_{g,1}/Y_{n+1}$ is surjective for $n\ge 2$, 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 $Y_n{\mathcal{I}\mathcal{C}}_{g,1}/Y_{n+2}$ for $n\ge 2$.

arXiv comment35 pages, 4 figures

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Cited by in corpus (2)

• A non-commutative Reidemeister-Turaev torsion of homology cylinders
• Torsion elements in the associated graded of the $Y$-filtration of the monoid of homology cylinders