An iterable surgery formula on involutive knot lattice homotopy

In ``Knots in lattice homology", Ozsv\'ath, Stipsicz, and Szab\'o showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology, and in previous work, I showed that knot lattice homology is the persistent homology of a doubly filtered space. Here I provide an iterable version of the surgery formula that, provided the initial knot lattice space with flip map, produces a space isomorphic as a doubly-filtered space to the corresponding knot lattice space for the dual knot with the corresponding flip map. If we include the involutive data for the original knot and ambient three-manifold, we can also produce the corresponding involutive data on the new knot lattice space without assuming that the original three-manifold is an $L$-space. I construct $\infty$-categories where these operations are functorial. Finally, I use the surgery formula to compute some examples of knot lattice spaces including for the regular fiber of $\Sigma(2,3,7)$ and for a knot in a three-manifold that is not given by an almost rational graph.