An extended symmetric union and its Alexander polynomial
Abstract
For prime knots K 1 and K 2 , we write K 1 ≥ K 2 if there is an epimorphism from the knot group of K 1 to that of K 2 which preserves the meridian. We construct a family of pairs of knots with K 1 ≥ K 2 such that an epimorphism maps the longitude of K 1 to the trivial element. This construction is regarded as an extension of a symmetric union with a single full twisted region. In particular, it extends a property of the Alexander polynomial of a symmetric union. We also exhibit that all but two of the knots up to ten crossings in the list of Kitano-Suzuki, which have an epimorphism mapping the longitude to the trivial element, arise from this construction.
Fields of Study & Subjects
Geometric and Algebraic Topology
Geometry and complex manifolds
Advanced Combinatorial Mathematics
Publication details
- Journal / source
- Topology and its Applications
- Publisher
- Elsevier BV
- Volume
- 390
- Pages
- 109893–109893
- Publication date
- 23 June 2026
- Language
- EN