A mnemonic for the Lipshitz-Ozsváth-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 $\widehat{HF}(S^3\setminus \mathring{\nu }(K))$ [arXiv:1604.03466] by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsváth, and Thurston [arXiv:0810.0687] calculating the bordered invariant of $S^3\setminus \mathring{\nu }(K)$ in terms of the knot Floer homology of $K$.

arXiv comment23 pages, 18 figures; v2: additional discussion and references; v3: This version has been accepted for publication at Algebraic & Geometric Topology

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• Linear independence of rationally slice knots
• Homology concordance and knot Floer homology