On the genera of symmetric unions of knots

TL;DR

The paper investigates symmetric unions of knots, focusing on even symmetric unions formed from a knot and its mirror. It proves a key identity for twisted Alexander polynomials that links the polynomial of an even symmetric union to that of its partial knot, yielding a genus bound g(K) \ge 2g(K_D) in this setting and providing partial answers to Jonathan Simon’s longstanding question. The authors also show that the genus inequality fails for skew-symmetric unions and demonstrate strong obstructions for certain multipart Montesinos knots, notably ruling out an even symmetric-union presentation for the Montesinos knot 11a_{201} via twisted Alexander-polynomial obstructions. Overall, the work connects symmetric-union constructions, epimorphisms of knot groups, and twisted invariants to constrain possible presentations and knot-genus relationships. These results offer practical criteria for ruling out symmetric-union presentations and guide future exploration of which ribbon knots admit such presentations.

Abstract

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.

Figures (8)

• Figure 1: Symmetric union $(D \cup D^\ast)(\infty, n_1, \dots, n_k)$ and its partial knot.
• Figure 2: Wirtinger generators $x, y, z, x^\ast, y^\ast, z^\ast,$ satisfy $yx = xz$ and $y^\ast x^\ast = x^\ast z^\ast$, and $\varphi_D(w_i)=w$ when $n_1\geq 0$.
• Figure 3: Generators for the knot groups of a symmetric union and a partial knot when $k=1$.
• Figure 4: Generators for the knot groups of a symmetric union and a partial knot.
• Figure 5: A skew-symmetric union presentation which is isotopic to a connected sum, and the associated partial knot.

Theorems & Definitions (49)

• Definition 1.2
• Proposition 1.4
• Proposition 1.5: Lam00
• Theorem 1.6
• Remark 1.7
• Corollary 1.8
• Proposition 1.9
• Remark 2.1
• proof : Proof of Theorem \ref{['thm:TAP']}
• Claim 3.1