arXiv 1505.016972015-05math.GTmath.AT

$S^1$上纤维化的三维流形中零同调纽结的有限型不变量

Finite type invariants of nullhomologous knots in 3-manifolds fibered over $S^1$ by counting graphs

Tadayuki Watanabe

备注：: 26 pages, 17 figures, v2: references updated, v3: main theorem weakened for n=2. arXiv admin note: text overlap with arXiv:1503.08735

摘要：

研究了闭3流形$M$中零同调纽结的有限型不变量，该流形定义为$M$中零同调纽结的合痕类所张成的向量空间$\mathscr{K}(M)$的某个降滤$\{\mathscr{K}_n(M)\}_{n\geq 0}$.滤过性$\{\mathscr{K}_n(M)\}_{n \geq 0}$是指在具有一个特殊叶片的$M$中对特殊类型的卡环进行手术。更准确地说，当$M$是纤维在$S^1$和$H_1(M)=\mathbb{Z}$，我们研究了有多远的自然手术映射，从空间的$\mathbb{Q}[t^{\pm 1}]$着色雅可比图的$S^1$的程度$n$到分级商$\mathscr{K}_n(M)/\mathscr{K}_{n+1}(M)$可以内射$n\leq 2$。为了做到这一点，我们构建了一个有限型不变量的零同源结$M$度2，这是一个类似的不变量在我们以前的论文arXiv：1503.08735，这是基于Lescop的建设$\mathbb{Z}$ -等变扰动不变量的3-流形。

英文摘要：

We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{\mathscr{K}_n(M)\}_{n\geq 0}$ of the vector space $\mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{\mathscr{K}_n(M)\}_{n \geq 0}$ is defined by surgeries on special kinds of claspers in $M$ having one special leaf. More precisely, when $M$ is fibered over $S^1$ and $H_1(M)=\mathbb{Z}$, we study how far the natural surgery map from the space of $\mathbb{Q}[t^{\pm 1}]$-colored Jacobi diagrams on $S^1$ of degree $n$ to the graded quotient $\mathscr{K}_n(M)/\mathscr{K}_{n+1}(M)$ can be injective for $n\leq 2$. To do this, we construct a finite type invariant of nullhomologous knots in $M$ up to degree 2 that is an analogue of the invariant given in our previous paper arXiv:1503.08735, which is based on Lescop's construction of $\mathbb{Z}$-equivariant perturbative invariant of 3-manifolds.

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英文摘要：

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