2014/05/09 by Yongqiang Liu, Laurenţiu Maxim, Laurentiu Maxim
Mathematics · #Alexander polynomial #Algebra over a field #Algebraic Geometry and Number Theory #Cohomology #Combinatorics #Geometric and Algebraic Topology #Hodge structure #Homotopy and Cohomology in Algebraic Topology #Hypersurface #Knot (papermaking) #Knot theory #Mathematical analysis #Mathematics #Polynomial #Pure mathematics #Torsion (gastropod) #math.AT #msc:14J70 #msc:32S20 #msc:32S25 #msc:32S55 #msc:32S60 #msc:57Q10
paper · pdf · doi:10.2140/agt.2015.15.2757
published as Algebr. Geom. Topol. 15 (2015) 2755-2785 · comments are very welcome
arxiv created 2014/05/09 · openalex publication_date 2015/11/12 · arxiv updated 2016/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Let f W C nC1 ! C be a polynomial that is transversal (or regular) at infinity. Let U D C nC1 n f 1 .0/ be the corresponding affine hypersurface complement. By using the peripheral complex associated to f , we give several estimates for the (infinite cyclic) Alexander polynomials of U induced by f , and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of U .