On the kernel of the surgery map restricted to the 1-loop part
On the kernel of the surgery map restricted to the 1-loop part
复制标题
仅限于1环部分的手术图核心
DOI：
10.1112/topo.12233
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发表时间：
2022
影响因子：
1.1
通讯作者：
Masaaki Suzuki
中科院分区：
文献类型：
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作者：
Yuta Nozaki;Masatoshi Sato;Masaaki Suzuki
每一个同调柱面都是由雅可比图通过卡环手术得到的。手术地图S：ANC→YnICg，1/YN+1$\Mathfrak{S}\冒号\数学{A}_n^c\右行Y_n\数学{i}\数学{C}_{g，1}/Y_(n+1)$是n⩾2$n\geqslant 2$的满射，其核与雅可比图的对称性密切相关。在取目标的某一商后，我们确定了限制在1圈部分的S{S}的核。在此基础上，我们进一步研究了n⩾2$n\geqslant 2$的交换群YnICg，1/Yn+2$Y_n\Mathcal{i}\Mathcal{C}_{g，1}/Y_(n+2)$.
Every homology cylinder is obtained from Jacobi diagrams by clasper surgery. The surgery map s:Anc→YnICg,1/Yn+1$\mathfrak {s}\colon \mathcal {A}_n^c \rightarrow Y_n\mathcal {I}\mathcal {C}_{g,1}/Y_{n+1}$ is surjective for n⩾2$n\geqslant 2$, and its kernel is closely related to the symmetry of Jacobi diagrams. We determine the kernel of s$\mathfrak {s}$ 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 YnICg,1/Yn+2$Y_n\mathcal {I}\mathcal {C}_{g,1}/Y_{n+2}$ for n⩾2$n\geqslant 2$.