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Strong corks derived from the Akbulut cork

Tateaki Mukohara

TL;DR

This work identifies and constructs strong corks by analyzing boundaries of corks arising from prior examples and by building new infinite families. It leverages two complementary frameworks: Heegaard Floer theory, via invariants $h_{\tau}(Y)$ and $h_{\iota\circ\tau}(Y)$, and instanton theory, via the invariant $r_s(Y,\tau)$, to certify non-extendability of boundary involutions over any smooth homology ball. The main contributions include proving that the boundaries of known corks are strong corks, showing that any nontrivial linear combination within natural families yields a strong cork, and constructing a broader family $(Z_{m,n},\tau)$ and another family $(Y_{m,n},\sigma)$ generalizing Akbulut-Yasui corks, with explicit proofs relying on equivariant cobordisms and spectral-invariant monotonicity. An appendix provides a distinguishing invariant (the Sakuma $\eta$-polynomial) to show the two strong involutions on certain knots are not Sakuma-equivalent, underscoring the richness of inequivalent equivariant structures. Overall, the results deepen the understanding of how cork twists produce exotic smooth structures and furnish new tools for producing and detecting absolutely exotic objects in 4-manifold topology.

Abstract

We prove that the boundaries of the corks introduced by Auckly, Kim, Melvin, and Ruberman in [AKMR14] and by Tange in [Tan16] are strong corks. Furthermore, we prove that any nontrivial linear combination of them yields a strong cork, and we construct a larger family of strong corks that generalizes them. These results rely on the instanton-theoretic invariant introduced by Alfieri, Dai, Mallick, and Taniguchi in [ADMT23].

Strong corks derived from the Akbulut cork

TL;DR

This work identifies and constructs strong corks by analyzing boundaries of corks arising from prior examples and by building new infinite families. It leverages two complementary frameworks: Heegaard Floer theory, via invariants and , and instanton theory, via the invariant , to certify non-extendability of boundary involutions over any smooth homology ball. The main contributions include proving that the boundaries of known corks are strong corks, showing that any nontrivial linear combination within natural families yields a strong cork, and constructing a broader family and another family generalizing Akbulut-Yasui corks, with explicit proofs relying on equivariant cobordisms and spectral-invariant monotonicity. An appendix provides a distinguishing invariant (the Sakuma -polynomial) to show the two strong involutions on certain knots are not Sakuma-equivalent, underscoring the richness of inequivalent equivariant structures. Overall, the results deepen the understanding of how cork twists produce exotic smooth structures and furnish new tools for producing and detecting absolutely exotic objects in 4-manifold topology.

Abstract

We prove that the boundaries of the corks introduced by Auckly, Kim, Melvin, and Ruberman in [AKMR14] and by Tange in [Tan16] are strong corks. Furthermore, we prove that any nontrivial linear combination of them yields a strong cork, and we construct a larger family of strong corks that generalizes them. These results rely on the instanton-theoretic invariant introduced by Alfieri, Dai, Mallick, and Taniguchi in [ADMT23].
Paper Structure (11 sections, 15 theorems, 33 equations, 26 figures)

This paper contains 11 sections, 15 theorems, 33 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Definitions and background
  4. Equivariant knots and equivariant surgery
  5. Strong corks
  6. detecting strong corks
  7. Heegaard-Floer-theoretic methods
  8. Proof of Theorem \ref{['thm_Y_m,n']}
  9. Instanton-theoretic methods
  10. Proof of Theorems \ref{['thm_Z_m,n']}, \ref{['thm_K_n']}, and \ref{['thm_Y_1,n']}
  11. appendix

Key Result

Theorem 1.2

For any $m,n\in\mathbb{N}$, let $(Z_{m,n},\tau)$ be the family of two-component link surgeries displayed in Figure Z_m,n. Equip $Z_{m,n}$ with the indicated involution $\tau$. Then $(Z_{m,n},\tau)$ is a strong cork. Furthermore, any nontrivial linear combination of elements in either $\{(Z_{m,n},\ta are strong corks. Here, for $a<0$, the notation $aZ$ denotes $|a|(-Z)$.

Figures (26)

  • Figure 1: Left: $(C_n,\tau)$. Here, the box labeled $\dfrac{-n+1}{2}$ represents $n-1$ negative half twists. Right: $(\overline{C}_n,\tau)$.
  • Figure 2: $(C(m),\tau)$.
  • Figure 3: $(Z_{m,n},\tau)$
  • Figure 4: $(Y_{m,n},\sigma)$.
  • Figure 5: $(Y_{1,n},\sigma)$.
  • ...and 21 more figures

Theorems & Definitions (34)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 24 more