The Rasmussen s-invariant and exotic 4-manifolds

TL;DR

This paper presents a simplified proof of the existence of exotic 4-manifolds by utilizing a generalized Rasmussen s-invariant for links and optimizing induction techniques, avoiding complex modules and lemmas.

Contribution

It introduces a streamlined, analysis-free approach to proving exotic 4-manifolds using generalized invariants and improved induction methods.

Findings

Existence of exotic 4-manifolds confirmed

Simplified proof avoiding skein lasagna modules

Enhanced computational efficiency through clever induction

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 Sto\v{s}i\'{c}'s induction scheme; this in particular allows us to avoid Ren and Willis's Comparison Lemma.