Miyazawa's Invariant, Lefschetz Numbers, and Seifert Solids

TL;DR

This paper links Miyazawa's 2-knot invariant to Lefschetz numbers in monopole Floer homology, providing new insights into 2-knots with Seifert solids and extending Floer homology techniques.

Contribution

It establishes a formula connecting Miyazawa's invariant to Lefschetz numbers and extends monopole Floer homology with Pin(2)-equivariant perturbations to integer coefficients.

Findings

Miyazawa's invariant expressed via Lefschetz number on monopole Floer homology.

Proves | ext{deg}|=1 for 2-knots with punctured L-spaces as Seifert solids.

Extends Floer homology with Pin(2)-equivariant perturbations to integer coefficients.

Abstract

We establish a formula expressing Miyazawa's 2-knot invariant $|\mathrm{deg}|$ in terms of the Lefschetz number of a map on ordinary (i.e., not real) monopole Floer homology. As an application, we deduce that $|\mathrm{deg}|=1$ for any 2-knot in $S^4$ which has a punctured $L$-space as a Seifert solid. In the course of the proof of the main theorem, we show how Francesco Lin's construction of monopole Floer homology with $\operatorname{Pin}(2)$-equivariant perturbations can be made to work with integer coefficients.