An extended symmetric union and its Alexander polynomial

TL;DR

This paper introduces an extended symmetric union construction for prime knots that preserves certain algebraic properties and demonstrates its relevance by classifying many small crossing knots with specific epimorphism properties.

Contribution

It presents a novel extension of symmetric unions that affects the Alexander polynomial and classifies most small crossing knots with particular epimorphisms.

Findings

Constructed a family of knot pairs with specific epimorphism properties.

Extended the property of the Alexander polynomial for symmetric unions.

Most small crossing knots with certain epimorphisms are explained by this construction.

Abstract

For prime knots $K_1$ and $K_2$, we write $K_1 \geq K_2$ 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 $K_1 \geq K_2$ 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.