Peculiar modules for 4-ended tangles

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.