Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by (Turaev 1988 Inventiones Math. 92 527-53), the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a connection between topological entanglement and quantum entanglement.