Seminar

The cosmetic surgery conjecture, posed by Cameron Gordon in 1990, is a uniqueness statement that (essentially) says a knot in an arbitrary 3-manifold is determined by its complement N. In the past three decades, this conjecture has been extensively studied, especially in the setting where the knot complement N embeds into the 3-sphere. Many different invariants of knots and 3-manifolds have been applied to this problem. After surveying some of these obstructions, I will describe some recent work (joint with Jessica Purcell and Saul Schleimer) that uses hyperbolic methods to reduce the cosmetic surgery conjecture for any particular cusped manifold N to a finite computer search.



Our work is implemented in code, available on Github, which takes as input a given cusped 3-manifold and tests it for cosmetic fillings. The talk will include a code demonstration. Among our computational results are that the cosmetic surgery conjecture holds for all knots in S^3 up to 19 crossings and all one-cusped manifolds in the SnapPy census up to 9 tetrahedra.