A mnemonic for the Lipshitz–Ozsváth–Thurston correspondence

Abstract

When k is a field, type D structures over the algebra kOEu; v=.uv/ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over kOEu; v=.uv/, can be viewed as a set of immersed curves. With this observation as a starting point, given a knot K in S 3 , we realize the immersed curve invariant c HF.S 3 X V .K// of Hanselman, Rasmussen and Watson by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsváth and Thurston calculating the bordered invariant of S 3 X V .K/ in terms of the knot Floer homology of K. 57K18, 57K31; 57R58Recent work interprets relative versions of homological invariants in terms of immersed curves, including Heegaard Floer homology for manifolds with torus boundary (see Hanselman, Rasmussen and Watson [4]) as well as link Floer homology (see Zibrowius [23]), singular instanton knot homology (see Hedden, Herald and Kirk [7]), and Khovanov homology (see Kotelskiy, Watson and Zibrowius [12]) for 4-ended tangles. In particular, Section 5 of [12] classifies type D structures over a quiver algebra associated with a surface with boundary in terms of immersed curves on this surface; compare Haiden, Katzarkov and Kontsevich [2] and Hanselman, Rasmussen and Watson [4]. Denoting a field by k, perhaps the simplest algebra to illustrate these classification results is R D kOEu; v=.uv/. This algebra arises as the path algebra of a quiver that is associated with the decorated surface shown in Figure 1. Work of Lekili and Polishchuk [13;14] describes the role of R, and its relationship with the twice-punctured disk, in the context of homological mirror symmetry; see in particular [14, Figures 1 and 2]. The algebra R equipped with the Alexander and ı gradings gr.u/ D . 1; 1/ and gr.v/ D .1; 1/ plays a central role in knot Floer homology; see Dai, Hom, Stoffregen and Truong [1], for instance.