An extended symmetric union with multiple tangle regions and its Alexander polynomial

TL;DR

This paper generalizes the extended symmetric union construction of knots to include multiple tangle regions, analyzing the resulting Alexander polynomial and group homomorphisms, thus expanding the understanding of knot invariants and structures.

Contribution

It introduces a generalized construction of extended symmetric unions with multiple tangle regions and characterizes their Alexander polynomial and group homomorphisms.

Findings

Alexander polynomial is the product of tangle numerator polynomials and the square of the partial knot's polynomial

Existence of a surjective homomorphism from the knot group of K to that of the partial knot

The construction broadens the class of knots with known algebraic properties

Abstract

The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element.