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Register a new userFinite type invariants of nullhomologous knots in 3-manifolds fibered over $S^1$ by counting graphs
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Authors
Tadayuki Watanabe
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We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration ${mathscr{K}_n(M)}_{ngeq 0}$ 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 $nleq 2$. 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 Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of 3-manifolds.
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We apply Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of AL-paths given in arXiv:1403.8030. We obtain an invariant $hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescops equivariant invariant $mathscr{Q}$ for 3-manifolds with $H_1=mathbb{Z}$ fibered over $S^1$. Moreover, surgery formulas of $hat{Z}_n$ and $mathscr{Q}$ for alternating sums of surgeries are given. This gives another proof of Lescops surgery formula of $mathscr{Q}$ for special kind of 3-manifolds and surgeries, which is simple in the sense that the formula is obtained easily by counting certain graphs in a 3-manifold.
For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the fiberwise gradient $xi$ of a fiberwise Morse function on $M$, we introduce the notion of amidakuji-like path (AL-path) on $M$. An AL-path is a piecewise smooth path on $M$ consisting of edges each of which is either a part of a critical locus of $xi$ or a flow line of $-xi$. Counting closed AL-paths with signs gives the Lefschetz zeta function of $M$. The moduli space of AL-paths on $M$ gives explicitly Lescops equivariant propagator, which can be used to define $mathbb{Z}$-equivariant version of Chern--Simons perturbation theory for $M$.
Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map ($mathbb{Z}pi$-homology equivalence) to a fixed 3-manifold $N$ and showed that there is a natural surjection from a space of $pi=pi_1N$-decorated graphs to the graded quotient of the filtration over $mathbb{Z}[frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$ the surjection of Garoufalidis--Levine is actually an isomorphism over $mathbb{Q}$. For the proof, we construct a perturbative invariant by applying Fukayas Morse homotopy theoretic construction to a local system of the quotient field of $mathbb{Q}pi$. The first invariant is an extension of the Casson invariant to $mathbb{Z}pi$-homology equivalences to the 3-torus. The results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with $b_1=3$. We also discuss some generalizations of the perturbative invariant for other target spaces $N$.
In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukayas Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozanskys space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.
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