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

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 [Formula: see text] 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 [Formula: see text], and there exists a surjective homomorphism from the knot group of K to that of [Formula: see text] which maps the longitude of K to the trivial element.