Abelian quotients of the Y –filtration on the homology cylinders via the LMO functor

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

“…Remark By Corollary 3.17, we have $$O(a_1,a_2,a_1)+O(a_2,a_1,a_2) \in \operatorname{Ker}\mathfrak {s}$$, which is proved in [15, Lemma 6.6(1)] in a different way. Now it is natural to ask whether $$\begin{equation*} O(a_1, \dots , a_{m-1}, a_m, a_{m-1}, \dots , a_1) + O(a_m, \dots , a_2, a_1, a_2, \dots , a_m) \in \operatorname{Ker}\mathfrak {s} \end{equation*}$$for $$m\geqslant 3$$.…”

“…Proof of Theorem Consider the exact sequence of abelian groups $$\begin{equation*} 0 \rightarrow Y_4\mathcal {I}\mathcal {C}/Y_5 \rightarrow Y_3\mathcal {I}\mathcal {C}/Y_5 \rightarrow Y_3\mathcal {I}\mathcal {C}/Y_4 \rightarrow 0. \end{equation*}$$By [15, Theorem 1.7], we have $$\operatorname{tor}(Y_3\mathcal {I}\mathcal {C}/Y_4) \cong (L_3 \oplus S^2(H))\otimes \mathbb {Z}/2\mathbb {Z}$$, where $$L_n$$ denotes the degree $$n$$ part of the free Lie algebra on $$H$$. Thus, the image of the set $$\begin{equation*} X= \lbrace T(a,b,c,b,a), T(b,c,a,c,b), T(a,b,b,b,a) \mid a \prec b \prec c\rbrace \cup \lbrace O(a^{\prime },b^{\prime },a^{\prime }) \mid a^{\prime }\preceq b^{\prime }\rbrace \end{equation*}$$under the map $$\mathfrak {s}$$ is a basis of …”

“…By focusing on 0‐loop Jacobi diagrams with three pairs of identical labels in Corollary 3.20, it suffices to see that the set $$\begin{equation*} \lbrace T(a,b,c,c,b,a), T(b,c,a,a,c,b) \mid a \prec b \prec c\rbrace \cup \lbrace O(a,b,b,b), O(a^{\prime },b^{\prime },b^{\prime },a^{\prime }) \mid a \prec b,\ a^{\prime }\preceq b^{\prime }\rbrace \end{equation*}$$extends to a basis of $$\mathcal {A}_4^c = \bigoplus _{l=0}^3 \mathcal {A}_{4,l}^c$$. This is shown by Example 6.4 and [15, Proposition 5.2].$$\Box$$…”

“…This conjecture asserts that the $$Y_n$$‐equivalence among homology cylinders is characterized by finite type invariants of degree at most $$n-1$$, which is one of the fundamental problems in quantum topology. The cases $$n=2,3$$ are known to be true (see [11, section 2.3]), and the case $$n=4$$ was solved in [15] as a consequence of the determination of the group $$Y_3\mathcal {I}\mathcal {C}/Y_{4}$$.…”

“…We use clasper calculus to study the kernel of $$\mathfrak {s}_{2k+1,k}$$. On the other hand, the image of $$\mathfrak {s}_{2k+1,k}$$ is investigated by the invariant $$\bar{z}_{2k+2}$$ introduced in [15]. As a byproduct of the proof of Theorem 6.9, we conclude that the $$(k+1)$$‐loop part $$\bar{z}_{2k+2,k+1}$$ of $$\bar{z}_{2k+2}$$ is non‐trivial in Corollary 6.7.…”