## Crossing Number of \$K_5\$ on the Tubular Neighborhood of a Trefoil Knot

The crossing number of a simple connected graph $$G$$ is the minimum number of edge crossings of $$G$$ over all drawing of $$G$$. It’s well known that on the sphere the crossing number of the complete graph $$K_5$$ is $$1$$ and on the torus it’s $$0$$.

What is the crossing number of $$K_5$$ on the tubular neighborhood of a trefoil knot?

My intuition is that it’s still $$0$$, but I’m having trouble showing this.