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As an example of the transitions between some of the eight geometries ofThurston, investigated before, we study the geometries supported by thecone-manifolds obtained by surgery on the trefoil knot with singular set thecore of the surgery. The geometric structures are explicitly constructed. Themost interesting phenomenon is the transition from $SL(2,\mathbb{R})$-geometryto $S^{3}$-geometry through Nil-geometry. A plot of the different geometries isgiven, in the spirit of the analogous plot of Thurston for the geometriessupported by surgeries on the figure-eight knot.

Geometric conemanifold structures on $\mathbb{T}_{p/q}$, the result of $p/q$ surgery in the left-handed trefoil knot $\mathbb{T}$