A curve-based survey of knot Floer homology and concordance invariants

Knot Floer homology associates to a knot in $S^3$ a bigraded chain complex over $\mathbb{F}[W,Z]$, from which many classical and concordance invariants can be extracted. Recent work shows that this algebraic object can equivalently be represented by a decorated immersed multicurve in a marked surface. This survey explains the immersed curve interpretation of knot Floer homology, aided by many examples, and shows how several invariants arising from the knot Floer complex can be extracted from the corresponding immersed curves. The decorated multicurve associated to a knot has a distinguished curve component ${\gamma }_0$ and a distinguished connected component ${\mathrm{\Gamma }}_0$, both of which are concordance invariants of the knot. We pay particular attention to these components and various numerical concordance invariants that can be extracted from them. We introduce new generalizations of the $V_s$ invariants, and we give a new curve-based description of the Upsilon invariant by showing that it is determined by generalized $V_s$ invariants.