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Torus surguries on knot traces

Kai Nakamura

TL;DR

$The paper develops a unified framework for studying knot traces via 4D torus surgeries, by interpreting Osoinach's annulus twists as torus surgeries on knot-trace exteriors $X_0(J_0)$ and then applying the rich theory of torus surgery to obtain exotic 4-manifolds, exotic traces, and potential counterexamples to the smooth 4D Poincaré conjecture. The main technical bridge, Theorem that annulus twists extend to torus surgeries, enables re-deriving Manolescu–Piccirillo homotopy-sphere standardness in this setting and extends to elliptic surfaces with the notion of $H$-slice. The work yields several novel outcomes: (i) infinite families of mutually exotic knot traces related by annulus twisting with concordance and identical shake-genus/genus-function data, (ii) exotic Stein traces and Legendrian realizations with identical boundaries, (iii) a hidden fishtail symmetry clarifying when annulus twists extend to trace diffeomorphisms and (iv) new potential candidates for SPC4 counterexamples built from torus surgery on Conway knot traces. Overall, the approach broadens the toolbox for constructing and distinguishing exotic 4-manifolds through knot-trace surgery and raises deep questions about sliceness, concordance, and the boundaries of Stein and symplectic fillings.$

Abstract

We initiate the study of torus surgeries on knot traces. Our key technical insight is realizing the annulus twisting construction of Osoinach as a torus surgery on a knot trace. We present several applications of this idea. We find exotic elliptic surfaces that can be realized as surgery on null-homologously embedded traces in a manner similar to that proposed by Manolescu and Piccirillo. Then we exhibit exotic traces with novel properties and improve upon the known geography for exotic Stein fillings. Finally, we construct new potential counterexamples to the smooth 4-dimensional Poincaré Conjecture.

Torus surguries on knot traces

TL;DR

X_0(J_0)H

Abstract

We initiate the study of torus surgeries on knot traces. Our key technical insight is realizing the annulus twisting construction of Osoinach as a torus surgery on a knot trace. We present several applications of this idea. We find exotic elliptic surfaces that can be realized as surgery on null-homologously embedded traces in a manner similar to that proposed by Manolescu and Piccirillo. Then we exhibit exotic traces with novel properties and improve upon the known geography for exotic Stein fillings. Finally, we construct new potential counterexamples to the smooth 4-dimensional Poincaré Conjecture.

Paper Structure

This paper contains 13 sections, 21 theorems, 1 equation, 12 figures.

Table of Contents

  1. Introduction
  2. Zero surgery homeomorphisms and exotic 4-manifolds
  3. Exotic traces with novel properties
  4. New potential counterexamples to SPC4
  5. Annulus twists as torus surgeries
  6. Manolescu-Piccirillo Homotopy Spheres
  7. Exotically annulus twisting
  8. Exotic Stein Traces
  9. A hidden fishtail symmetry
  10. A second family
  11. Motivation and Further Questions
  12. Slice Akbulut-Kirby Conjecture
  13. Knot traces

Key Result

Theorem 1.1

An annulus twist homeomorphism $\phi_k:S^3_0(J_0) \rightarrow S^3_0(J_k)$ can be realized $4$-dimensionally as a torus surgery of $X_0(J_0)$ resulting in $X_0(J_k)$.

Figures (12)

  • Figure 1: Annulus presentation of $J_0$ and its $k$ fold annulus twist $J_k$
  • Figure 2: Annulus twisting as a torus surgery
  • Figure 3: Realizing slice trace surgery as an unknotted torus surgery
  • Figure 4: (a)Twist knot $\kappa_k$ (b) Surgery description of $\kappa_k$ as $\kappa_0$ and $-1/k$ surgery on the red knot $C$
  • Figure 5: Kirby diagrams of our knot traces
  • ...and 7 more figures

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Theorem 1.3 of trace_emb_zero
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 34 more