[Paper Review] Connected sums of knots do not admit purely cosmetic surgeries

Ran Tao|arXiv (Cornell University)|Sep 11, 2019

Geometric and Algebraic Topology11 references7 citations

TL;DR

This paper confirms the Cosmetic Surgery Conjecture for composite knots by analyzing their JSJ-decompositions, proving that no non-trivial connected sum of knots admits purely cosmetic Dehn surgeries—i.e., distinct surgery slopes cannot yield homeomorphic oriented 3-manifolds. The argument relies on distinguishing JSJ structures under orientation-preserving homeomorphisms and resolving contradictions in fiber-slopes and gluing patterns for slopes ±1, ±2, and ±1/q.

ABSTRACT

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

Motivation & Objective

• To verify the Cosmetic Surgery Conjecture for composite knots, i.e., connected sums of non-trivial knots in S³.
• To extend previous results on cable knots to the broader class of composite knots using JSJ-topological techniques.
• To rule out purely cosmetic surgeries by analyzing the topological invariants of Dehn surgered manifolds.
• To resolve the remaining difficult cases of surgeries with slopes ±1 and ±2 through detailed analysis of JSJ-piece symmetries and gluing patterns.

Proposed method

• Apply the JSJ-decomposition theorem to the knot complement and its Dehn surgered manifolds to decompose them into Seifert fibered and atoroidal pieces.
• Use the uniqueness of JSJ-tori up to isotopy to compare the JSJ-structures of S³_r(J) and S³_s(J) under an orientation-preserving homeomorphism h.
• Employ obstruction theorems from Heegaard Floer homology (Theorem 1.3) to restrict possible cosmetic slope pairs to {±1}, {±2}, or {±1/q} for q > 1.
• Analyze fiber-slopes and gluing patterns on boundary tori: show that homeomorphisms preserve slopes and fiber directions, leading to contradictions when fiber-slopes do not match.
• Use Lemma 4.9 to show that the union of certain JSJ-pieces (Z ∪ S₁) forms a knot complement with meridional fiber-slope.
• Contradict the existence of a homeomorphism h by comparing meridian/longitude gluing in original and image manifolds under h, especially for hyperbolic and cable space components.

Experimental results

Research questions

• RQ1 Can a composite knot in S³ admit purely cosmetic Dehn surgeries with distinct slopes?
• RQ2 Do the JSJ-decompositions of S³_r(J) and S³_s(J) remain homeomorphic under an orientation-preserving homeomorphism h when r ≠ s?
• RQ3 What topological obstructions arise when comparing fiber-slopes and gluing patterns across JSJ-pieces under such a homeomorphism?
• RQ4 How do the constraints from Theorem 1.3 (on slope pairs {±1}, {±2}, {±1/q}) interact with the JSJ-structure of composite knots?
• RQ5 Can the contradiction in fiber-slope intersections (e.g., 1 vs. q ≥ 2) be used to rule out homeomorphisms for slopes ±1?

Key findings

• For composite knots J, no orientation-preserving homeomorphism exists between S³_r(J) and S³_s(J) when r ≠ s, confirming the Cosmetic Surgery Conjecture for this class.
• The case of slopes {±1} is ruled out via contradiction: the fiber-slope of a cable space component must be meridional in one manifold but have intersection number q ≥ 2 with meridians in the image, violating slope preservation.
• For {±2} surgeries, the conjecture is excluded via Theorem 1.3, as the genus of a composite knot with ≥3 prime summands exceeds 2.
• The union of JSJ-pieces Z ∪ S₁ forms a knot complement with meridional fiber-slope, a key invariant preserved under homeomorphism.
• When Y is hyperbolic, the homeomorphism h would require meridians of one piece to be glued to longitudes of another, contradicting the original meridian-to-meridian gluing pattern.
• The fiber-slope of a cable space component at its boundary is not meridional in the image under h, contradicting the original meridional fiber-slope in the domain.