Purely cosmetic surgery conjecture

Cut a thick tube around a knotted loop out of the three-sphere, then glue a solid torus into the opening. Can two different choices of which boundary curve becomes the meridian of the inserted torus produce the same oriented three-dimensional space? This operation is Dehn surgery, and that curve’s direction on the boundary torus is its slope. For example, one loop around the meridian and a loop going once in each meridian and longitude direction are different slopes. The conjecture says their oriented results cannot agree for any two distinct slopes on the same nontrivial knot. Agreement must preserve orientation; reversing it is a separate question.[1] [2]

Can two different boundary directions disappear as a distinction after the same knot exterior is filled? Slopes are unoriented isotopy classes of essential simple closed curves on its boundary torus. Using a meridian–longitude basis, $(1,0)$ and $(1,1)$ are primitive and unequal even after identifying a pair with its negative; this checks slope distinctness, not the resulting manifolds. For every nontrivial knot $K\subset S^3$ and distinct slopes $r\ne s$, the target is $S^3_K(r)\not\cong_+S^3_K(s)$. The plus denotes orientation-preserving homeomorphism. Orientation reversal, several torus boundaries and the unknot are outside the quantifiers.[1] [2]

The published quantum obstruction paper verifies the target for knots with at most seventeen crossings. The September preprint reports it when the Jones polynomial is nontrivial and its span, the gap between its largest and smallest exponents, is at most eleven. These are bounded or restricted families, not the universal result; the crossing threshold is that paper’s result, not a claim about the current best threshold. Reported examples using several boundary tori or orientation reversal do not supply a counterexample to this single-knot oriented target.[1] [2] [3]

The quantum obstruction verifies a specified crossing-bounded family, and the low-span preprint requires both a nontrivial Jones polynomial and span at most eleven. Those restrictions leave arbitrary knots uncovered by these statements. Ren’s constructions concern multiple cusps or chirally cosmetic comparisons, and the one-cusp purely cosmetic target is not an example they construct. A cited report of another crossing threshold in that preprint is not independently certified as a current-best result here.[1] [2] [3]

Surgery describes a closed three-dimensional space by a knot and a gluing instruction. Distinct instructions need not generally be easy to distinguish by inspecting the resulting space. This conjecture tests how much of the instruction survives in the oriented topology when the original nontrivial knot is fixed. It would prevent the slope from being lost in that fixed-knot description, connecting a concrete construction to the classification of closed spaces. It does not assert uniqueness of every surgery description using different knots.[1] [2]

The oriented filling type is information about a completed three-manifold, while the slope is information about how it was constructed. Injectivity of the filling type for a fixed nontrivial knot would connect those descriptions and limit redundancy in surgery parametrizations. The cited quantum obstruction uses Witten–Reshetikhin–Turaev and colored Jones data to forbid coincidences, giving an algebraic route to a topological classification question. Such an obstruction can distinguish eligible cases without being a complete invariant or proving uniqueness across different knot exteriors.[1] [2]