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

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.

Figures (14)

• Figure 1: the diagram of a knot $K$
• Figure 2: two special rational tangles
• Figure 3: labels in the diagram of $L$
• Figure 4: labels aroud crossing $d_i$ and $d_i^\ast$
• Figure 5: the diagram of $K_1$ in the case $n=1$

Theorems & Definitions (4)

• Theorem 1.1
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['maintheorem']}(1)