Branched covers of twist-roll spun knots and turned twisted tori

TL;DR

The paper studies 2-fold branched covers of 4-manifolds branched along twist-roll spun knots and turned-torus constructions. By lifting knot-surgery operations to the 2-fold cover and analyzing lifted tori and curves, it proves invariance relations: $\Sigma_{2}(\tau^{m}\rho^{n}(K))\cong\Sigma_{2}(\tau^{m+4}\rho^{n}(K))$ and $\Sigma_{2}(\tau^{m}\rho^{n}(K)\#P_{\pm})\cong\Sigma_{2}(\tau^{m+2}\rho^{n}(K)\#P_{\pm})$, leading to diffeomorphism classifications in several families. Consequently, Miyazawa’s family of homotopy $\mathbb{CP}^{2}$s are each diffeomorphic to $\mathbb{CP}^{2}$, the double branched covers of odd-twisted turned tori are $S^{2}\times S^{2}$, and the homotopy 4-spheres constructed by Juhász–Powell are standard $S^{4}$. The paper also extends these methods to twist-spun and turned-torus complements, providing a unified framework for understanding how twist and turn operations influence branched-cover manifolds.

Abstract

We prove that the double branched cover of a twist-roll spun knot in $S^4$ is smoothly preserved when four twists are added, and that the double branched cover of a twist-roll spun knot connected sum with a trivial projective plane is preserved after two twists are added. As a consequence, we conclude that the members of a family of homotopy $\mathbb{CP}^2$s recently constructed by Miyazawa are each diffeomorphic to $\mathbb{CP}^2$. We also apply our techniques to show that the double branched covers of odd-twisted turned tori are all diffeomorphic to $S^2 \times S^2$, and show that a family of homotopy 4-spheres constructed by Juhász and Powell are all diffeomorphic to $S^4$.

Figures (7)

• Figure 1: A smoothly unknotted 2-sphere $S\subset S^4$ and an unknotted torus $T$ that bounds a solid torus centered about a curve on $S$. On $T$ we indicate a curve $C_1$ that bounds a framed disk in the complement of $T$ that intersects $S$ once and a dual curve $C_2$ that bounds a framed disk in the complement of $S\cup T$.
• Figure 2: Litherland's description of the 2-knot $\tau^m\rho^n (K)$, which is obtained by identifying the ends of $(B^3, \mathring{K}) \times [0,1]$ under the diffeomorphism $\phi_{m,n}$, and gluing in $(S^2 \times D^2, \partial \mathring{K} \times D^2)$.
• Figure 3: The Hopf tangle $\mathring{H}$, whose spin yields the link $S\cup T$.
• Figure 4: Left: the unknotted torus $\widetilde{T}$ in $S^4$. From left to right, we exhibit an ambient isotopy of $S^4$ taking $\widetilde{T}$ to itself setwise but effecting two Dehn twists about $\widetilde{C}_1$. Reversing the direction of the isotopy achieves Dehn twists of the opposite sign.
• Figure 5: Top left: The unknotted torus $\widetilde{T}$ in $\mp\mathbb{CP}^2$, represented via a banded unlink diagram as in hugheskimmiller:isotopies. We perform a slide of a band over a 2-handle and then a swim taking the 2-handle attaching circle through the other band. This describes an ambient isotopy taking $\widetilde{T}$ to $\widetilde{T}$ setwise but achieving a Dehn twist about $\widetilde{C}_1$. Reversing the isotopy achieves a Dehn twist of the opposite sign.

Theorems & Definitions (27)

• Theorem 1.1
• Corollary 1.2
• proof : Proof of Corollary \ref{['cor:pretzel']}
• Corollary 1.5
• proof : Proof of Corollary \ref{['cor:involution']}
• Proposition 1.6
• Corollary 1.7
• Theorem 1.9
• Proposition 1.10
• Theorem 1.11