Infinitely many standard trisection diagrams for Gluck twisting

TL;DR

The paper proves that the trisection diagram for the Gluck twist on the spun $(p+1,p)$-torus knot is standard for all $p\ge 2$, addressing a question of Gay–Meier. It constructs the diagram $\mathcal{D}_p$ via Meier's spinning method from a doubly-pointed diagram of $(S^3,t(3,2))$, applies a sequence of Dehn twists and handle slides, and then destabilizes and reduces the diagram using the seesaw lemma to the stabilization of the genus 0 trisection of $S^4$. The main result provides a systematic, inductive diagrammatic approach to show standardness in a broad family of Gluck twists, suggesting that certain Gluck-twisted 4-manifolds diffeomorphic to $S^4$ admit standard trisections. The work connects to broader questions about the Waldhausen-type conjecture for $S^4$ trisections and highlights techniques for constructing and simplifying Gluck-twisted trisection diagrams.

Abstract

Gay and Meier asked if a trisection diagram for the Gluck twist on a spun or twist-spun 2-knot in $S^4$ obtained by a certain method is standard. In this paper, we show that the trisection diagram for the Gluck twist on the spun $(p+1,p)$-torus knot is standard, where $p$ is any integer greater than or equal to 2.

Figures (21)

• Figure 1: The genus 1 trisection diagrams of $S^4$.
• Figure 2: A doubly-pointed Heegaard diagram of $(S^3, t(3,2))$ and the curves $\delta_1$ and $\delta_2$.
• Figure 3: An arced relative trisection diagram of $S^4-S(t(3,2))$ and the curves $\delta_1^{\beta}$, $\delta_2^{\beta}$, $\delta_1^{\gamma}$ and $\delta_2^{\gamma}$.
• Figure 4:
• Figure 5:

Theorems & Definitions (20)

• Theorem : Theorem \ref{['thm:main']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Theorem 4.1
• Remark 4.2