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Doubly slicing knots and embedding 3-manifolds in 4-manifolds

Se-Goo Kim, Taehee Kim

TL;DR

This work extends the double slice theory by defining the $X$-double slice genus $g_{ds}^X(K)$ for knots in $S^3$ relative to a simply connected 4-manifold $X$, and develops a network of obstructions based on $H_1$-embeddings and $\rho^{(2)}$-invariants. The authors prove that $g_{ds}^X(K)$ can be arbitrarily large for any fixed $X$, even when $K$ is algebraically doubly slice or when all prime-power branched covers embed into $S^4$ and Casson–Gordon obstructions vanish. They relate genus bounds to $H_1$-embedding obstructions, extend the stabilizing and superslice notions to $X$-contexts, and define the $X$-double stabilizing number and $X$-superslice genus, proving they can be made arbitrarily large and establishing several tight inequalities. Together, these results illuminate the intricate landscape of how 3-manifolds bound, embed, and interact with 4-manifold topology through generalized slice and stabilizing invariants, with potential implications for understanding knotted surfaces in higher dimensions.

Abstract

For a knot $K$ in the 3-sphere and a simply connected closed 4-manifold $X$, we define the $X$-double slice genus of $K$, extending the notion from the case when $X$ is the 4-sphere. We show that for each integer $n$, there exists an algebraically doubly slice and ribbon knot $K$ whose $X$-double slice genus is greater than $n$. Our arguments use new $L^2$-signature obstructions to embedding closed 3-manifolds with infinite cyclic first homology into closed 4-manifolds with infinite cyclic fundamental group, in a way that preserves first homology. We also extend the concept of the superslice genus of a knot to simply connected 4-manifolds and show that there exist doubly slice knots whose generalized superslice genera are arbitrarily large. Furthermore, we define the double stabilizing number of a knot, extending the stabilizing number introduced by Conway and Nagel, and show that this invariant can also be arbitrarily large.

Doubly slicing knots and embedding 3-manifolds in 4-manifolds

TL;DR

This work extends the double slice theory by defining the -double slice genus for knots in relative to a simply connected 4-manifold , and develops a network of obstructions based on -embeddings and -invariants. The authors prove that can be arbitrarily large for any fixed , even when is algebraically doubly slice or when all prime-power branched covers embed into and Casson–Gordon obstructions vanish. They relate genus bounds to -embedding obstructions, extend the stabilizing and superslice notions to -contexts, and define the -double stabilizing number and -superslice genus, proving they can be made arbitrarily large and establishing several tight inequalities. Together, these results illuminate the intricate landscape of how 3-manifolds bound, embed, and interact with 4-manifold topology through generalized slice and stabilizing invariants, with potential implications for understanding knotted surfaces in higher dimensions.

Abstract

For a knot in the 3-sphere and a simply connected closed 4-manifold , we define the -double slice genus of , extending the notion from the case when is the 4-sphere. We show that for each integer , there exists an algebraically doubly slice and ribbon knot whose -double slice genus is greater than . Our arguments use new -signature obstructions to embedding closed 3-manifolds with infinite cyclic first homology into closed 4-manifolds with infinite cyclic fundamental group, in a way that preserves first homology. We also extend the concept of the superslice genus of a knot to simply connected 4-manifolds and show that there exist doubly slice knots whose generalized superslice genera are arbitrarily large. Furthermore, we define the double stabilizing number of a knot, extending the stabilizing number introduced by Conway and Nagel, and show that this invariant can also be arbitrarily large.
Paper Structure (10 sections, 15 theorems, 58 equations, 1 figure)

This paper contains 10 sections, 15 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cheeger--Gromov--von Neumann $\rho^{(2)}$-invariants
  3. $H_1$--embedding of 3-manifolds in 4-manifolds
  4. The double slice genus and $H_1$--embeddings
  5. The $X$--double slice genus of a knot
  6. The double slice genus and knots whose prime power fold branched cyclic covers embed in $S^4$
  7. The double stabilizing number of a knot
  8. The double stabilizing number
  9. The $X$--double stabilizing number
  10. The $X$--superslice genus of a knot

Key Result

Theorem 1

Let $M$ be a closed 3-manifold with $H_1(M)\cong \mathbb{Z}=\langle t\rangle$ and $W$ be a closed 4-manifold with $\pi_1(W)\cong \mathbb{Z}$. If $f\colon M\longrightarrow W$ is an $H_1$--embedding, then there exist 4-manifolds $W_1$ and $W_2$ for which the following hold: let $r=b_2(W)$ and $\Lambda

Figures (1)

  • Figure 1: The knot $R$

Theorems & Definitions (32)

  • Remark 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.2
  • Theorem 4
  • Theorem 5
  • Remark 1.3
  • Theorem 6
  • Theorem 7
  • ...and 22 more