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

TL;DR

This paper investigates finite type invariants of nullhomologous knots in fibered 3-manifolds over S^1, exploring the injectivity of a surgery map and constructing invariants up to degree 2 using graph counting techniques.

Contribution

It introduces a new framework for finite type invariants in fibered 3-manifolds and constructs explicit invariants up to degree 2, extending previous perturbative methods.

Findings

The surgery map's injectivity is analyzed for degree n ≤ 2.

A finite type invariant up to degree 2 is constructed.

The work relates diagrammatic invariants to clasper surgeries in fibered 3-manifolds.

Abstract

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.