Real link Floer homology

TL;DR

This paper introduces real link Floer homology for certain symmetric links in 3-manifolds, generalizing existing theories, with combinatorial descriptions, structural insights, and computational results for small knots.

Contribution

It defines a new real link Floer homology theory for strongly invertible links, extending previous work, and provides combinatorial tools and computational data.

Findings

Development of a combinatorial description via real grid diagrams in S^3

Investigation of structural properties of the new theory

Computational analysis of real grid homology for over 50 small knots

Abstract

In this paper, we define real link Floer homology for strongly invertible and doubly periodic links in closed real $3$-manifolds with connected fixed sets, which generalizes real Heegaard Floer homology and real sutured Heegaard Floer homology. We give a combinatorial description of the theory in $S^3$ via real grid diagrams and use it to investigate structural properties of the theory as well as properties of strongly invertible knots. A computer implementation was written by Zhenkun Li. An appendix including real grid homology for 50+ small knots is made jointly by Zhenkun Li and the author, from which we observe several interesting phenomenon.