On elementary invariants of genus one knots and Seifert surfaces
arXiv:2304.04707 · doi:10.1007/978-3-031-81414-3_20
Abstract
This elementary article introduces easy-to-manage invariants of genus one knots in homology 3-spheres. To prove their invariance, we investigate properties of an invariant of 3-dimensional genus two homology handlebodies called the Alexander form. The Alexander form of a 3-manifold E with boundary contains all Reidemeister torsions of link exteriors obtained by attaching two-handles along the boundary of E. It is a useful tool for studying Alexander polynomials and Reidemeister torsions. We extract invariants of genus one Seifert surfaces from the Alexander form of their exteriors.
54 pages, 12 figures, minor revision, mainly typographical