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The Rasmussen s-invariant and exotic 4-manifolds

Gheehyun Nahm

Abstract

We give a short exposition of Ren and Willis's analysis-free proof of the existence of exotic compact, orientable 4-manifolds. There are two distinguishing features of our exposition. First, we avoid skein lasagna modules; we use Beliakova and Wehrli's generalization of Rasmussen's s-invariant to links in $S^{3}$ directly. Second, we reduce the complexity of the computations by choosing clever induction hypotheses in Stošić's induction scheme; this in particular allows us to avoid Ren and Willis's Comparison Lemma.

The Rasmussen s-invariant and exotic 4-manifolds

Abstract

We give a short exposition of Ren and Willis's analysis-free proof of the existence of exotic compact, orientable 4-manifolds. There are two distinguishing features of our exposition. First, we avoid skein lasagna modules; we use Beliakova and Wehrli's generalization of Rasmussen's s-invariant to links in directly. Second, we reduce the complexity of the computations by choosing clever induction hypotheses in Stošić's induction scheme; this in particular allows us to avoid Ren and Willis's Comparison Lemma.
Paper Structure (1 section, 7 theorems, 11 equations)

This paper contains 1 section, 7 theorems, 11 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

The homeomorphic knot traces $X_{1}(5_{2})$ and $X_{1}(P(-3,3,8))$ are not diffeomorphic.

Theorems & Definitions (17)

  • Theorem 1
  • Definition 2
  • Theorem 3: ren2024khovanovv3
  • proof : Proof of Theorem \ref{['thm:mainthm']} assuming Theorem \ref{['thm:sinv']}
  • Lemma 5: Main lemma
  • proof : Proof of Theorem \ref{['thm:sinv']} assuming Lemma \ref{['lem:band-map']}
  • Definition 6: Auxiliary links for the induction scheme
  • Definition 7: Renormalization for the induction scheme
  • Lemma 8: Homological gradings of Lee generators, ren2024khovanovv3
  • proof
  • ...and 7 more