On symmetries of peculiar modules; or, $\delta$-graded link Floer homology is mutation invariant

TL;DR

This paper studies the symmetry properties of peculiar modules in Heegaard Floer theory, demonstrating that Conway mutation preserves the delta-graded link Floer homology, thus revealing invariance under certain tangle transformations.

Contribution

It provides a comprehensive analysis of the symmetry of peculiar modules and proves mutation invariance of delta-graded link Floer homology.

Findings

Conway mutation preserves the hat flavor of delta-graded Heegaard Floer link invariants.

Almost complete classification of components of peculiar modules of tangles.

Symmetry properties of peculiar modules are characterized extensively.

Abstract

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar modules of tangles. As a main application, we show that Conway mutation preserves the hat flavour of the relatively $\delta$-graded Heegaard Floer theory of links.