sway-n; 2 Senses

Sense Number 1: having political or social influence over something

Commentary: SWAY[+state][+social][+influence]

That part of Europe is no longer under Russia's sway.
His decisions were clearly under the sway of upper management.
The coach holds a lot of sway with his team members.

WordNet 3.0 Sense Numbers: 1

Sense Number 2: tilting or rocking side to side

Commentary: SWAY[+action][+physical][+motion][+tilting]

The audience enjoyed the sway of the dancers' hips.
The sway of the branches in the wind was a peaceful sound.

WordNet 3.0 Sense Numbers: 2