New curved Kakeya estimates

We give new lower bounds for the Hausdorff dimension of Kakeya sets built from various families of curves in ${\mathbb{R}}^3$, going beyond what the polynomial partitioning method has so-far achieved. We do this by combining Wolff’s classical hairbrush argument with a new incidence bound for 3-parameter families of curves which satisfy conditions we call coniness and twistiness. Our main argument builds off a technique of Katz, Wu, and Zahl used in the study of ${\mathrm{S}\mathrm{L}}_2$-Kakeya sets.