# Involutive bordered Floer homology

## TL;DR

This paper extends involutive Heegaard Floer homology to bordered 3-manifolds, providing algorithms for computation, exploring mapping class group actions, and applying these to surgery formulas and knot invariants.

## Contribution

It introduces a bordered Floer homology extension for involutive HF-hat, enabling broader computations and applications in 3-manifold topology.

## Key findings

- Algorithm for involutive HF-hat of general 3-manifolds
- Proof of surgery exact triangle for involutive HF-hat
- Computed HFI-hat for branched double covers of 10-crossing knots

## Abstract

We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact triangle and compute HFI-hat of the branched double covers of all 10-crossing knots.