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List of knot theory topics

From Wikipedia, the free encyclopedia

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Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

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History

Knots, links, braids

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Notation used in knot theory:

General knot types

General types of links:

Tangles

Braids

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Operations

Elementary treatment using polygonal curves

  • elementary move (R1 move, R2 move, R3 move)
  • R-equivalent
  • delta-equivalent

Invariants and properties

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Mathematical problems

Theorems

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