Gluck twist and unknotting of satellite $2$-knots

TL;DR

This paper investigates how Gluck twists affect satellite 2-knots in 4-manifolds, showing they often do not change the knot type and providing new examples of unknottable 2-knots through connected sums with a real projective plane.

Contribution

It demonstrates that Gluck twists do not alter certain satellite 2-knots and introduces a new description leading to infinitely many unknottable 2-knots via connected sum with a real projective plane.

Findings

Gluck twists do not change the diffeomorphism type of certain satellite 2-knots.

New description of satellite 2-knots enables construction of infinitely many unknottable 2-knots.

Provided examples of 2-knots unknotted by connected sum with a real projective plane.

Abstract

In this paper, we show that the Gluck twist of certain satellite $2$-knots in a $4$-manifold do not change the diffeomorphism type in three different ways: one is directly from the definition of the satellite $2$-knot, and the other two are by finding an equivalent description of the satellite $2$-knot. Furthermore, using the new description, we gave infinite number of new examples of $2$-knots which are unknotted by connected summing a single standard real projective plane.