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 $K$.

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

References in corpus (4)

• Floer homology and knot complements
• Cabling in terms of immersed curves
• Immersed curves in Khovanov homology
• Algebras with matchings and knot Floer homology

Cited by in corpus (2)

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