An iterable surgery formula on involutive knot lattice homotopy

TL;DR

This paper develops an iterative surgery formula for involutive knot lattice homology, enabling the construction of dual knot spaces with preserved involutive structures, and applies it to compute examples in complex three-manifolds.

Contribution

It introduces an iterable surgery formula for involutive knot lattice homology, extending previous work to include involutive data without the L-space restriction.

Findings

Constructed functorial $ abla$-categories for the surgery operations.

Produced involutive knot lattice spaces for specific examples.

Extended the surgery formula to non-L-space three-manifolds.

Abstract

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.