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

Abstract

We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration $\{{\mathcal{K}}_n(M){\}}_{n\ge 0}$ of the vector space $\mathcal{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration $\{{\mathcal{K}}_n(M){\}}_{n\ge 0}$ 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 ${\mathcal{K}}_n(M)/{\mathcal{K}}_{n+1}(M)$ can be injective for $n\le 2$. 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.

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

References in corpus (4)

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• A universal equivariant finite type knot invariant defined from configuration space integrals
• Morse theory and Lescop's equivariant propagator for 3-manifolds with $b_1=1$ fibered over $S^1$
• An invariant of fiberwise Morse functions on surface bundle over $S^1$ by counting graphs