Math 2.0

Manifolds and cell complexes Definition

Torus knot

Builds on KnotTorus

Definition

Let pp and qq be coprime integers. The (p,q)(p, q)-torus knot T(p,q)T(p, q) is the image of the map f ⁣:ℝ/2πℤ→ℝ3f \colon \R / 2\pi\Z \to \R^3, f(t)=((2+cos⁡qt)cos⁡pt, (2+cos⁡qt)sin⁡pt, −sin⁡qt).f(t) = \big( (2 + \cos qt) \cos pt, \ (2 + \cos qt) \sin pt, \ -\sin qt \big). A knot is a torus knot if it is isotopic to T(p,q)T(p, q) for some coprime p,qp, q with ∣p∣≥2\abs{p} \ge 2 and ∣q∣≥2\abs{q} \ge 2.

Remarks

The image lies on the standard torus obtained by revolving the circle (x−2)2+z2=1(x - 2)^2 + z^2 = 1 about the zz-axis. The map ff is injective because pp and qq are coprime. Example: T(2,3)T(2, 3) is the trefoil knot; here ∣p∣,∣q∣≥2\abs{p}, \abs{q} \ge 2, so it is a torus knot in the sense above. Non-example: T(1,q)T(1, q) is isotopic to the unknot, so the bound ∣p∣≥2\abs{p} \ge 2 fails and it is not counted as a torus knot. T(p,q)T(p, q) is isotopic to T(q,p)T(q, p), and T(p,−q)T(p, -q) is the mirror image of T(p,q)T(p, q). For p,q≥2p, q \ge 2 the Seifert genus of T(p,q)T(p, q) is (p−1)(q−1)/2(p-1)(q-1)/2.

Reading path

To understand this page from the ground up, read these 65 pages in order. The foundations come first.

  1. Element definition
  2. Knot concept
  3. Proposition concept
  4. Real numbers object
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Set-builder notation concept
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. Union of a family of sets definition
  21. Uniqueness quantifier definition
  22. Binary relation definition
  23. Universal set definition
  24. Topological space definition
  25. Reflexive relation definition
  26. Symmetric relation definition
  27. Transitive relation definition
  28. Complement definition
  29. Function definition
  30. Basis of a topology definition
  31. Subspace topology definition
  32. Equivalence relation definition
  33. Metric space definition
  34. Product topology definition
  35. Preimage of a set definition
  36. Codomain definition
  37. Function composition definition
  38. Injective function definition
  39. Open ball definition
  40. Equivalence class definition
  41. Surjective function definition
  42. Metric topology definition
  43. Quotient set definition
  44. Bijective function definition
  45. Open set definition
  46. Quotient topology definition
  47. Inverse function definition
  48. Hausdorff space definition
  49. Open cover definition
  50. Closed set definition
  51. Continuous map of topological spaces definition
  52. Compact space definition
  53. Complement of an open set is closed statement
  54. Homeomorphism definition
  55. Universal property of the quotient topology statement
  56. Continuity via closed sets statement
  57. Compact set definition
  58. Compactness via open covers statement
  59. Compact subsets of a Hausdorff space are closed statement
  60. Closed subsets of compact sets are compact statement
  61. Continuous image of a compact set is compact statement
  62. Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
  63. Circle as a quotient of the real line example
  64. Torus example
  65. Torus knot definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Torus_knot
About this page
TagMD1QFAEFF0
URIm2://MD1QFAEFF0
Kinddefinition
WritingW0 stub
MSC57K10
DefinesTorus knot
UsesKnot, Torus
Changed2 November 2026

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