An extended symmetric union and its Alexander polynomial

TL;DR

The paper addresses the problem of constructing knot pairs $(K,\hat{K})$ with a meridian-preserving epimorphism $\varphi:G(K)\to G(\hat{K})$ that sends the longitude to the trivial element, extending symmetric unions to include a single full-twisted region. It develops an extended symmetric-union framework in which $\Delta_K(t)=\Delta_{K'}(t)\left(\Delta_{\hat{K}}(t)\right)^2$ and proves a longitude-trivial epimorphism via an explicit $\varphi$, supported by a Conway-polynomial factorization $\nabla_K(z)=\nabla_{N(\tilde{T})}(z)\left(\nabla_{\hat{K}}(z)\right)^2$. The results yield infinite families of Montesinos and 2-bridge knots with $K\geq\hat{K}$ and longitude mapped to the trivial element, including infinitely many non-fibered examples, while providing explicit knots up to 10 crossings and highlighting two exceptions ($10_{82},10_{87}$) not arising from the construction. The work unifies symmetric-union properties with polynomial and group-epimorphism structures, offering a broad, constructive toolkit for understanding when knot groups admit longitude-trivial, meridian-preserving epimorphisms.

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.

Figures (13)

• Figure 1: the diagram of a knot $K$
• Figure 2: $K=N(\tilde{T}+T_0)$
• Figure 3: $L_0$, $L_1$, $L_2$
• Figure 4: arcs of $D$, $D^\ast$, $T_0$
• Figure 5: arcs of $D$, $D^\ast$, and $T$, the oriented diagram of $K$

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Proposition 3.1
• Lemma 3.2