A note on the involutive invariants of splices

TL;DR

This paper investigates the involutive Floer homology of spliced knots and their mirrors, revealing that many such 3-manifolds have trivial involutive Floer homology and related gauge invariants.

Contribution

It demonstrates that splices of knots with their mirrors often have locally trivial involutive Floer homology, expanding understanding of their gauge theoretic invariants.

Findings

Spliced knots with mirrors have locally trivial involutive Floer homology.

Many gauge theoretic invariants vanish on these spliced 3-manifolds.

The results apply to various families of splices beyond the initial examples.

Abstract

A natural family of potentially 2-torsion elements in the integer homology cobordism group consists of splices of knots with their mirrors. We show that such 3-manifolds have locally trivial involutive Floer homology. We show some related families of splices also have locally trivial involutive Floer homology. Our arguments show that many gauge theoretic invariants also vanish on these 3-manifolds.