2015/03/30 by Tadayuki Watanabe, Watanabe, Tadayuki
Computer Science · Mathematics · #57M27 #57R57 #58D29 #58E05 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GT #msc:57M27 #msc:57R57 #msc:58D29 #msc:58E05
paper · pdf · doi:10.48550/arxiv.1503.08735
44 pages, 16 figures, v2, corrected errors
openalex publication_date 2015/03/30 · arxiv created 2015/05/07 · arxiv updated 2015/05/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We apply Lescop's construction of 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 Zn of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over S1, 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 Lescop's equivariant invariant \mathscrQ for 3-manifolds with H1=ℤ fibered over S1. Moreover, surgery formulas of Zn and \mathscrQ for alternating sums of surgeries are given. This gives another proof of Lescop's surgery formula of \mathscrQ 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.