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

TL;DR

This paper explores the geometric structures of cone-manifolds resulting from surgery on the left-handed trefoil knot, highlighting a transition from $SL(2,\mathbb{R})$-geometry to $S^3$-geometry via Nil-geometry, with explicit constructions and visualizations.

Contribution

It provides explicit constructions of geometric structures on cone-manifolds from trefoil knot surgeries and reveals a novel transition between Thurston's geometries.

Findings

Transition from $SL(2,\mathbb{R})$ to $S^3$-geometry through Nil-geometry.

Explicit geometric structures for cone-manifolds from trefoil surgeries.

Visualization of geometry transitions similar to Thurston's plot for figure-eight knot.

Abstract

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