A note on PL-disks and rationally slice knots

TL;DR

This paper constructs examples of knots in 3-manifolds that bound smooth disks in rational homology balls but not in integer homology balls, highlighting subtle differences in knot slicing properties.

Contribution

It introduces infinitely many manifold-knot pairs demonstrating new distinctions in PL-disk bounding within different homology contexts using involutive Heegaard Floer homology.

Findings

Existence of manifold-knot pairs with specific bounding properties

Demonstration of knots bounding disks in rational but not integer homology balls

Application of involutive Heegaard Floer homology to knot theory

Abstract

We give infinitely many examples of manifold-knot pairs (Y, J) such that Y bounds an integer homology ball, J does not bound a non-locally-flat PL-disk in any integer homology ball, but J does bound a smoothly embedded disk in a rational homology ball. The proof relies on formal properties of involutive Heegaard Floer homology.