The salient crossings of a crown diagram

TL;DR

This paper introduces a method using crown diagrams and integer assignments to distinguish smooth structures on certain 4-manifolds that share the same Seibus- ter-Witten invariants, showing they are not isotopic.

Contribution

It provides a new approach to differentiate smooth structures on 4-manifolds via crown diagrams and integer assignments, extending previous invariants.

Findings

Distinguishes non-isotopic smooth structures on Fintushel-Stern knot surgery 4-manifolds

Shows these manifolds are not isotopic despite having the same Seiberg-Witten invariants

Introduces a novel diagrammatic technique for analyzing 4-manifold smooth structures

Abstract

A crown diagram of a smooth, closed oriented 4-manifold can be thought of as the projection of a link in the product of a closed surface and the circle, with chords in the circle direction connecting the strands of each crossing. This paper uses a straightforward assignment of integers to these chords to show that the smooth structures on the topological 4-manifold underlying a pair of Fintushel-Stern knot surgery 4-manifolds which are known to have the same Seiberg-Witten invariant are not isotopic. A natural question is to determine if the argument may be strengthened to show these manifolds are not diffeomorphic.