Consider a Conway two-sphere S intersecting a knot K in 4 points, and thus decomposing the knot into two 4-ended tangles, T and T'. We will first interpret Khovanov homology Kh(K) as Lagrangian Floer homology of a pair of specifically constructed immersed curves Kh(T) and BN(T') on the dividing 4-punctured sphere S. Next, motivated by tangle-replacement questions in knot theory, we will describe a recently obtained structural result concerning the curve invariant Kh(T), which severely restricts the types of curves that may appear as tangle invariants. The proof relies on the matrix factorization framework of Khovanov-Rozansky, as well as the homological mirror symmetry statement for the 3-punctured sphere. This is joint work with Liam Watson and Claudius Zibrowius.
This talk was part of the conference "Perspectives on quantum link homology theories", see https://cbz20.raspberryip.com/Perspectives-2021