Peculiar modules for 4-ended tangles

TL;DR

This paper introduces a new invariant called the peculiar module for 4-ended tangles, proves a glueing formula relating it to link Floer homology, classifies these modules via immersed curves, and explores applications including tangle detection and skein relations.

Contribution

It develops the theory of peculiar modules for 4-ended tangles, establishes a glueing formula, classifies modules via immersed curves, and applies these results to tangle detection and Floer homology invariants.

Findings

Peculiar modules recover link Floer homology through a glueing formula.

Classification of peculiar modules using immersed curves on the 4-punctured sphere.

Peculiar modules detect rational tangles and preserve certain Floer homology under mutation.

Abstract

With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which recovers link Floer homology $\operatorname{\widehat{HFL}}$. Moreover, we classify peculiar modules in terms of immersed curves on the 4-punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson, we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4-punctured sphere. We then study some applications: firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2-stranded pretzel tangles $T_{2n,-(2m+1)}$ for $n,m>0$ using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles $T_{2n,-(2m+1)}$ preserves $\delta$-graded, and for some orientations even bigraded link Floer homology.