Higher Steenrod squares for Khovanov homology

Categorical lifting of the Jones polynomial: a survey

“…If the Khovanov homology has a sufficiently simple form then these, in turn, determine the stable homotopy type. The operation Sq 1 is just the Bockstein homomorphism, and one can give an explicit formula for the Steenrod squaring operation Sq 2 [110], and more complicated formulas for all Steenrod squares [29]. The operation Sq 2 is enough to determine the stable homotopy type for all knots up to 14 crossings and, in fact, some pairs of knots with isomorphic Khovanov homologies are distinguished by their Steenrod squares [159].…”

Categorical lifting of the Jones polynomial: a survey

“…If the Khovanov homology has a sufficiently simple form, then these, in turn, determine the stable homotopy type. The operation Sq 1 is just the Bockstein homomorphism, and one can give an explicit formula for the Steenrod squaring operation Sq 2 [109] and more complicated formulas for all Steenrod squares [30]. The operation Sq 2 is enough to determine the stable homotopy type for all knots up to 14 crossings and, in fact, some pairs of knots with isomorphic Khovanov homologies are distinguished by their Steenrod squares [158].…”