A mnemonic for the Lipshitz-Ozsv\'ath-Thurston correspondence

Abstract

When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over $\mathbf{k}[u,v]/(uv)$, 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 $\widehat{\mathit{HF}}(S^3 \smallsetminus \mathring{\nu}(K))$ [arXiv:1604.03466] by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsv\'ath, and Thurston [arXiv:0810.0687] calculating the bordered invariant of $S^3 \smallsetminus \mathring{\nu}(K)$ in terms of the knot Floer homology of $K$.