Finite type invariants of nullhomologous knots in 3-manifolds fibered over S¹ by counting graphs

2015/05/07 by Watanabe, Tadayuki
#57M27 #57R57 #58D29 #58E05 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1505.01697

Abstract

We study finite type invariants of nullhomologous knots in a closed 3-manifold M defined in terms of certain descending filtration \\mathscrK_n(M)\_{n≥ 0} of the vector space \mathscrK(M) spanned by isotopy classes of nullhomologous knots in M. The filtration \\mathscrK_n(M)\_{n ≥ 0} is defined by surgeries on special kinds of claspers in M having one special leaf. More precisely, when M is fibered over S¹ and H₁(M)=ℤ, we study how far the natural surgery map from the space of ℚ[t^{± 1}]-colored Jacobi diagrams on S¹ of degree n to the graded quotient \mathscrK_n(M)/\mathscrK_n+1(M) can be injective for n≤ 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 ℤ-equivariant perturbative invariant of 3-manifolds.

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