Peculiar modules for 4‐ended tangles. Issue 1 (19th August 2019) - Zibrowius, Claudius
Abstract: With a 4‐ended tangle T, we associate a Heegaard Floer invariant CFT ∂ ( T ), the peculiar module of T . Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology H F L ̂ . 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 (Preprint, 2016, arXiv:1604.03466v2), 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 2 n, − ( 2 m + 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 2 n, − ( 2 m + 1 ) preserves δ ‐graded, and for some orientations even bigraded link Floer homology.