Non-Trivial Steenrod Squares on Prime, Hyperbolic and Satellite Knots

TL;DR

This paper demonstrates the existence of prime, hyperbolic, and satellite knots with non-trivial Steenrod operations on their Khovanov homology, answering a question posed by Lipshitz and Sarkar.

Contribution

It establishes the presence of non-trivial Steenrod operations on various classes of knots' Khovanov homology, expanding understanding of their algebraic structures.

Findings

Non-trivial Steenrod operations on prime knots

Existence of hyperbolic knots with non-trivial Steenrod operations

Presence of satellite knots with non-trivial Steenrod operations

Abstract

We show that there are prime knots so that the Steenrod operations of Lipshitz and Sarkar arXiv:1204.5776 are non trivial on their Khovanov homology. This answers a question posed by Lipshitz and Sarkar in their paper arXiv:1709.03602. We then go on to show that there are hyperbolic and satellite knots so that the Steenrod operations are non trivial on their Khovanov homology.