{ "id": "epd3ntvzkc", "slug": "dehn-surgery", "title": "Dehn surgery", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.geometry", "label": "Geometry and topology", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.geometry" }, { "id": "physical.math.geometry.topology_geometric", "label": "Geometric topology and low-dimensional topology", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.geometry.topology_geometric" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math", "label": "Non-geographic · 1800 to 1945: Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" } ] } ], "excerpt": "Dehn surgery is a construction in 3-manifold topology that removes a solid torus and reglues it along a slope p/q, yielding every closed orientable 3-manifold.", "snippet": "Dehn surgery is a construction in 3-manifold topology that removes a solid torus and reglues it along a slope p/q, yielding every closed orientable 3-manifold.", "node": "physical.math.geometry.topology_geometric", "markdown": "# Dehn surgery\n\nDehn surgery is a cut-and-paste construction in 3-manifold topology: from a 3-manifold M one removes the interior of a solid torus N and glues a solid torus \\( N_{1} \\) back in its place by a homeomorphism h of the boundary torus, giving a new manifold \\( M_{1} = (M \\setminus \\operatorname{Int} N) \\cup_{h} N_{1} \\).[1](https://encyclopediaofmath.org/wiki/Dehn_surgery) The gluing is parametrized by a rational slope p/q, and the Lickorish–Wallace theorem guarantees that every closed orientable 3-manifold arises from the 3-sphere by surgery on a link.[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) • [3](https://doi.org/10.2307/1970373)\n\n| Key fact | Statement |\n| --- | --- |\n| Surgery data | A knot K in M and a slope p/q ∈ Q ∪ {∞}; the pair (p, q) and (−p, −q) define the same surgery.[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) • [4](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf) |\n| Homology | The filling K(p/q) has \\( H_{1} \\cong \\mathbb{Z}/p\\mathbb{Z} \\), generated by the oriented meridian.[4](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf) |\n| Unknot | \\( p/q \\)-surgery on the unknot gives the lens space \\( L(p, q) \\); \\( 1/n \\)-surgery gives \\( S^{3} \\).[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) • [4](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf) |\n| Poincaré sphere | +1-surgery on the right-handed trefoil yields the Poincaré homology sphere, as does −1-surgery on the left-handed trefoil.[4](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf) • [2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) |\n| Universality | Every closed orientable 3-manifold is obtained from S³ by Dehn surgery on a link (Lickorish–Wallace).[3](https://doi.org/10.2307/1970373) • [2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) |\n| Equivalence of descriptions | Two integral surgery descriptions give the same manifold exactly when they differ by Kirby moves.[5](https://doi.org/10.1007/bf01406222) |\n| Exceptional slopes | The figure-eight knot exterior has exactly 10 exceptional slopes: {−4, −3, −2, −1, 0, 1, 2, 3, 4, ∞}.[6](https://people.maths.ox.ac.uk/lackenby/claysurgery2.pdf) |\n\n## How it works\n\nThe boundary of a tubular neighborhood ν(K) of a knot carries a canonical basis. The meridian µ bounds a disk in ν(K), and the longitude λ is nullhomologous in the complement Y \\ ν(K); identifying ∂ν(K) with ℝ²/ℤ² turns gluing maps into integer matrices, and the orientation-preserving ones form \\( \\mathrm{SL}^{+}(2, \\mathbb{Z}) \\).[7](https://math.berkeley.edu/~nm.eagles/notes/LKS_NME.pdf) Surgery removes Int ν(K) and glues a solid torus V so that the meridian of \\( \\partial V \\) maps to \\( p\\mu + q\\lambda \\), with (p, q) coprime; this is rational \\( p/q \\) surgery.[1](https://encyclopediaofmath.org/wiki/Dehn_surgery) • [7](https://math.berkeley.edu/~nm.eagles/notes/LKS_NME.pdf) The outcome is uniquely determined by the image of the meridian alone, independent of where the longitude goes.[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf)\n\n## How it is done\n\nIn practice a surgery is drawn as a framed link diagram in \\( S^{3} \\): each component carries an integer coefficient, meaning \\( p/q \\)-surgery with that integer value. Kirby's 1978 calculus decides when two such diagrams give the same manifold: two integral surgery diagrams are diffeomorphic if and only if they are related by handle slides and blow-ups and blow-downs (adding or deleting a \\( \\pm 1 \\)-framed unknotted component), with blow-ups and blow-downs changing the framings of components that link the new one by \\( \\pm \\) its linking number with it.[5](https://doi.org/10.1007/bf01406222) • [1](https://encyclopediaofmath.org/wiki/Dehn_surgery) Handle slides remain necessary in general; blow-ups and blow-downs alone do not suffice.[8](https://etnyre.math.gatech.edu/class/8803Fall21/V.%20More%20On%20Dehn%20Surgery.pdf) Sliding a component with coefficient p/q over one with coefficient n changes the coefficient to \\( r' = p/q + n - 2\\,\\mathrm{lk}(K, K') \\), where lk is the linking number.[8](https://etnyre.math.gatech.edu/class/8803Fall21/V.%20More%20On%20Dehn%20Surgery.pdf)\n\nRational coefficients are handled by Rolfsen twists: two rational surgery diagrams give diffeomorphic manifolds exactly when they are related by Rolfsen twists, a rational extension of Kirby's theorem published by Dale Rolfsen in 1984.[8](https://etnyre.math.gatech.edu/class/8803Fall21/V.%20More%20On%20Dehn%20Surgery.pdf) • [9](https://doi.org/10.2140/pjm.1984.110.377)\n\n## Origin\n\nThe construction grew out of early attempts to build homology 3-spheres, manifolds with the homology of \\( S^{3} \\) that are not \\( S^{3} \\), by identifying the boundaries of two knot exteriors; an attempt of this kind could not show the resulting manifold differed from S³, and the cut-and-paste operation itself appears in the literature shortly afterward.[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) The modern centrality of the method rests on the theorem that every closed orientable 3-manifold is surgery on a link in \\( S^{3} \\), building on the twist theorem that every orientation-preserving surface homeomorphism is a product of Dehn twists; a companion approach starts from the fact that every closed orientable 3-manifold bounds a smooth compact orientable 4-manifold.[3](https://doi.org/10.2307/1970373) [Robion Kirby](https://www.edgepedia.org/robion-kirby)'s 1978 Inventiones mathematicae calculus for framed links then made descriptions computable,[5](https://doi.org/10.1007/bf01406222) and Dale Rolfsen extended it to rational coefficients in his 1984 Pacific Journal of Mathematics paper.[9](https://doi.org/10.2140/pjm.1984.110.377) The 1980s added the main rigidity results: the Cyclic Surgery Theorem of Marc Culler and colleagues (1987, [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics)),[10](https://doi.org/10.2307/1971311) Gordon and Luecke's 1989 Journal of the American Mathematical Society theorem that knots are determined by their complements,[11](https://doi.org/10.1090/s0894-0347-1989-0965210-7) and [David Gabai](https://www.edgepedia.org/david-gabai)'s 1987 Journal of Differential Geometry proof that no surgery on a non-trivial knot in \\( S^{3} \\) gives \\( S^{2} \\times S^{1} \\).[12](https://doi.org/10.4310/jdg/1214441488)\n\n## Variants\n\n**Integral versus rational.** On S³ the preferred longitude bounds a surface in the complement, so the attaching curve has the form \\( l = m^{p} l_{0}^{q} \\) with p, q coprime, and the surgery is determined by \\( r = p/q \\); the surgery is integer exactly when r is an integer.[1](https://encyclopediaofmath.org/wiki/Dehn_surgery) Integral surgery means all coefficients in a link description are integers.\n\n**Dehn filling.** For a manifold with torus boundary components, Thurston's notes write the filling as M(α₁, β₁), …, (αₖ, βₖ), gluing solid tori so the i-th meridian maps to \\( \\alpha_{i} a_{i} + \\beta_{i} b_{i} \\); the coefficient \\( \\infty \\) means nothing is glued in, leaving a cusp.[13](https://www.math.unl.edu/~mbrittenham2/classwk/990s08/public/thurston.notes.pdf/4a.pdf)\n\n**±1 surgery and handles.** Lickorish's version of the universality theorem uses only \\( \\pm 1 \\) surgeries along a link in \\( S^{3} \\).[4](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf) Integral surgeries correspond to attaching 4-dimensional 2-handles to \\( Y \\times [0, 1] \\); in contact geometry, Legendrian surgery is topologically −1-surgery with the 2-handle framed \\( \\mathrm{tb}(K) - 1 \\).[7](https://math.berkeley.edu/~nm.eagles/notes/LKS_NME.pdf)\n\n## Applications\n\nThe hyperbolic Dehn surgery theorem states that a finite-volume orientable hyperbolic 3-manifold has a finite set E of exceptional slopes such that the filling M(s₁, …, sₙ) is hyperbolic whenever each sᵢ ∉ E, with the filling cores becoming arbitrarily short geodesics.[14](https://people.maths.ox.ac.uk/lackenby/dehn-surgery-icerm-28dec2020.pdf) Hyperbolic deformations of the complete structure are parametrized near (∞, …, ∞) by the filling coefficients: coprime integer coefficients give a nonsingular hyperbolic manifold, rational ones a cone manifold.[15](https://www.numdam.org/article/AST_2001__272__179_0.pdf) Quantitative strengthenings make the theorem effective. Craig Hodgson and Steven Kerckhoff proved that if the normalized length of the filling slope (geodesic length divided by \\( \\sqrt{\\text{torus area}} \\)) is at least 7.515, the filling is hyperbolic, with a larger threshold for multiple cusps, and that at most 60 Dehn fillings of a one-cusped manifold fail to be hyperbolic, or at most 114 surgery curves per torus in the multi-cusped case.[16](https://doi.org/10.4007/annals.2005.162.367) A finiteness theorem implies a manifold arises as \\( p/q \\) surgery on a hyperbolic knot in \\( S^{3} \\) with \\( |q| > 11 \\) for only finitely many knots and slopes.[14](https://people.maths.ox.ac.uk/lackenby/dehn-surgery-icerm-28dec2020.pdf) Surgery rules also organize [Wilson loop](https://www.edgepedia.org/wilson-loop) observables in quantum Chern–Simons field theory, the setting of Witten–Reshetikhin–Turaev-type invariants.[17](https://doi.org/10.1016/0550-3213%2892%2990037-c)\n\n## Limitations and alternatives\n\n**Exceptional slopes.** Thurston proved the figure-eight knot exterior has exactly 10 exceptional slopes, \\( \\{-4, -3, -2, -1, 0, 1, 2, 3, 4, \\infty\\} \\), and Gordon conjectured 10 as a universal maximum.[6](https://people.maths.ox.ac.uk/lackenby/claysurgery2.pdf) Lackenby and Meyerhoff (2015) proved a one-cusped finite-volume orientable hyperbolic manifold has at most 10 exceptional slopes, with the distance between any two at most 8; whether the figure-eight exterior is the unique worst case remains open, though Agol showed only finitely many manifolds attain 10.[14](https://people.maths.ox.ac.uk/lackenby/dehn-surgery-icerm-28dec2020.pdf) Reducible fillings are rare: if \\( M(s_{1}) \\) and \\( M(s_{2}) \\) are both reducible, the slopes have intersection number 1, so at most 3 slopes give reducible fillings.[6](https://people.maths.ox.ac.uk/lackenby/claysurgery2.pdf) The 6-theorem says a slope of length \\( L(s) > 6 \\) gives an irreducible, atoroidal, non-Seifert-fibred filling with infinite word-hyperbolic fundamental group; the Gromov–Thurston 2π-theorem gives negatively curved metrics when \\( L(s) > 2\\pi \\), and at most 48 slopes have \\( L(s) \\le 2\\pi \\).[6](https://people.maths.ox.ac.uk/lackenby/claysurgery2.pdf)\n\n**Rigidity of the surgery map.** The Cyclic Surgery Theorem states that for a knot in \\( S^{3} \\) other than a [torus knot](https://www.edgepedia.org/torus-knot), two cyclic surgeries must satisfy \\( |p_{1}q_{2} - p_{2}q_{1}| \\le 1 \\).[2](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf) • [10](https://doi.org/10.2307/1971311) No non-trivial surgery on a non-trivial knot gives \\( S^{3} \\) (Gordon–Luecke) or \\( S^{2} \\times S^{1} \\) (Gabai).[11](https://doi.org/10.1090/s0894-0347-1989-0965210-7) • [12](https://doi.org/10.4310/jdg/1214441488) A cosmetic surgery pair is two distinct slopes s, s′ with a homeomorphism N(s) → N(s′), purely cosmetic if orientation-preserving and chirally cosmetic if orientation-reversing; Gordon's conjecture denies purely cosmetic surgeries on manifolds with incompressible torus boundary.[18](https://doi.org/10.20382/jocg.v16i1a19)\n\nAs an alternative construction, integral surgery on a link is equivalent to a handle decomposition of a 4-manifold with 2-handles attached, so surgery presentations and handle diagrams describe the same objects from two directions.[1](https://encyclopediaofmath.org/wiki/Dehn_surgery)\n\n## References\n\n1. [Dehn surgery, Encyclopedia of Mathematics (S.V. Matveev)](https://encyclopediaofmath.org/wiki/Dehn_surgery)\n2. [Dehn surgery on knots: survey (Tetsuya Ito / Kimihiko Motegi, Differential Topology 22, University of Tsukuba, March 2022)](http://www.math.tsukuba.ac.jp/~tange/diftop22/Dehn_surgery_survey.pdf)\n3. [W. B. R. Lickorish (1962). A Representation of Orientable Combinatorial 3-Manifolds. Annals of Mathematics.](https://doi.org/10.2307/1970373)\n4. [Dehn surgery (expository notes, Siddhartha Gadgil)](https://math.iisc.ac.in/%7Egadgil/expos/surgery.pdf)\n5. [Robion Kirby (1978). A calculus for framed links inS 3. Inventiones mathematicae.](https://doi.org/10.1007/bf01406222)\n6. [The Dehn Surgery Problem (Marc Lackenby, Clay Mathematics Institute lecture slides)](https://people.maths.ox.ac.uk/lackenby/claysurgery2.pdf)\n7. [Legendrian Surgeries (seminar notes, UC Berkeley)](https://math.berkeley.edu/~nm.eagles/notes/LKS_NME.pdf)\n8. [More on Dehn Surgery (course notes, John Etnyre, Georgia Tech)](https://etnyre.math.gatech.edu/class/8803Fall21/V.%20More%20On%20Dehn%20Surgery.pdf)\n9. [Dale Rolfsen (1984). Rational surgery calculus: extension of Kirby’s theorem. Pacific Journal of Mathematics.](https://doi.org/10.2140/pjm.1984.110.377)\n10. [Marc Culler and colleagues (1987). Dehn Surgery on Knots. Annals of Mathematics.](https://doi.org/10.2307/1971311)\n11. [C. McA. Gordon, J. Luecke (1989). Knots are determined by their complements. Journal of the American Mathematical Society.](https://doi.org/10.1090/s0894-0347-1989-0965210-7)\n12. [David Gabai (1987). Foliations and the topology of 3-manifolds. III. Journal of Differential Geometry.](https://doi.org/10.4310/jdg/1214441488)\n13. [Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4: Hyperbolic Dehn surgery](https://www.math.unl.edu/~mbrittenham2/classwk/990s08/public/thurston.notes.pdf/4a.pdf)\n14. [Hyperbolic methods in Dehn surgery (Marc Lackenby, ICERM lecture notes, 2019/2020)](https://people.maths.ox.ac.uk/lackenby/dehn-surgery-icerm-28dec2020.pdf)\n15. [Appendix B: Thurston's hyperbolic Dehn filling theorem (Astérisque 272)](https://www.numdam.org/article/AST_2001__272__179_0.pdf)\n16. [Craig Hodgson, Steven Kerckhoff (2005). Universal bounds for hyperbolic Dehn surgery. Annals of Mathematics.](https://doi.org/10.4007/annals.2005.162.367)\n17. [Surgery rules in quantum Chern-Simons field theory (Nuclear Physics B, 1992)](https://doi.org/10.1016/0550-3213%2892%2990037-c)\n18. [Excluding cosmetic surgeries on hyperbolic 3-manifolds (Futer, Purcell, Schleimer, Journal of Computational Geometry, 2025)](https://doi.org/10.20382/jocg.v16i1a19)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Geometric topology and low-dimensional topology*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://math.berkeley.edu/~nm.eagles/notes/LKS_NME.pdf", "https://www.math.unl.edu/~mbrittenham2/classwk/990s08/public/thurston.notes.pdf/4a.pdf" ], "url": "https://www.edgepedia.org/dehn-surgery", "markdown_url": "https://www.edgepedia.org/dehn-surgery.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Dehn surgery\", Edgepedia (EdgeChat), https://www.edgepedia.org/dehn-surgery. Edgepedia Community License 1.0.", "credit_md": "\"[Dehn surgery](https://www.edgepedia.org/dehn-surgery)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/dehn-surgery](https://www.edgepedia.org/dehn-surgery). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Dehn surgery\", Edgepedia (EdgeChat), https://www.edgepedia.org/dehn-surgery. Edgepedia Community License 1.0.", "speakable": "Dehn surgery is a construction in 3-manifold topology that removes a solid torus and reglues it along a slope p/q, yielding every closed orientable 3-manifold." }