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, R<sup>3</sup>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R<sup>3</sup> 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.

History

  • Knot (mathematics) gives a general introduction to the concept of a knot.
  • Two classes of knots: torus knots and pretzel knots
  • Cinquefoil knot also known as a (5,&nbsp;2) torus knot.
  • Figure-eight knot (mathematics) the only 4-crossing knot
  • Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots
  • Perko pair, two entries in a knot table that were later shown to be identical.
  • Stevedore knot (mathematics), a prime knot with crossing number&nbsp;6
  • Three-twist knot is the twist knot with three-half twists, also known as the&nbsp;5<sub>2</sub>&nbsp;knot.
  • Trefoil knot A knot with crossing number&nbsp;3
  • Unknot
  • Knot complement, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere.

Notation used in knot theory:

  • Conway notation
  • Dowker–Thistlethwaite notation (DT notation)
  • Gauss code (see also Gauss diagrams)
  • continued fraction

General knot types

  • 2-bridge knot
  • Alternating knot; a knot that can be represented by an alternating diagram (i.e. the crossing alternate over and under as one traverses the knot).
  • Berge knot a class of knots related to Lens space surgeries and defined in terms of their properties with respect to a genus 2 Heegaard surface.
  • Cable knot, see Satellite knot
  • Chiral knot is knot which is not equivalent to its mirror image.
  • Double torus knot, a knot that can be embedded in a double torus (a genus 2 surface).
  • Fibered knot
  • Framed knot
  • Invertible knot
  • Prime knot
  • Legendrian knot are knots embedded in <math>\mathbb R^3</math> tangent to the standard contact structure.
  • Lissajous knot
  • Ribbon knot
  • Satellite knot
  • Slice knot
  • Torus knot
  • Transverse knot
  • Twist knot
  • Virtual knot
  • Wild knot
  • Borromean rings, the simplest Brunnian link
  • Brunnian link, a set of links which become trivial if one loop is removed
  • Hopf link, the simplest non-trivial link
  • Solomon's knot, a two-ring link with four crossings.
  • Whitehead link, a twisted loop linked with an untwisted loop.
  • Unlink

General types of links:

  • Algebraic link
  • Hyperbolic link
  • Pretzel link
  • Split link
  • String link

Tangles

  • Tangle (mathematics)
  • Algebraic tangle
  • Tangle diagram
  • Tangle product
  • Tangle rotation
  • Tangle sum
  • Inverse of a tangle
  • Rational tangle
  • Tangle denominator closure
  • Tangle numerator closure
  • Reciprocal tangle

Braids

  • Braid theory
  • Braid group

Operations

  • Band sum
  • Flype
  • Fox n-coloring
  • Tricolorability
  • Knot sum
  • Reidemeister move

Elementary treatment using polygonal curves

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

Invariants and properties

  • Knot invariant is an invariant defined on knots which is invariant under ambient isotopies of the knot.
  • Finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots
  • Knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
  • Alexander polynomial and the associated Alexander matrix; The first knot polynomial (1923). Sometimes called the Alexander–Conway polynomial
  • Bracket polynomial is a polynomial invariant of framed links. Related to the Jones polynomial. Also known as the Kauffman bracket.
  • Conway polynomial uses Skein relations.
  • Homfly polynomial or HOMFLYPT polynomial.
  • Jones polynomial assigns a Laurent polynomial in the variable t<sup>1/2</sup> to the knot or link.
  • Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.
  • Arf invariant of a knot
  • Average crossing number
  • Bridge number
  • Crosscap number
  • Crossing number
  • Hyperbolic volume
  • Kontsevich invariant
  • Linking number
  • Milnor invariants
  • Racks and quandles and Biquandle
  • Ropelength
  • Seifert surface
  • Self-linking number
  • Signature of a knot
  • Skein relation
  • Slice genus
  • Tunnel number, the number of arcs that must be added to make the knot complement a handlebody
  • Writhe

Mathematical problems

  • Berge conjecture
  • Birman&ndash;Wenzl algebra
  • Clasper (mathematics)
  • Eilenberg&ndash;Mazur swindle
  • Fáry–Milnor theorem
  • Gordon–Luecke theorem
  • Khovanov homology
  • Knot group
  • Knot tabulation
  • Knotless embedding
  • Linkless embedding
  • Link concordance
  • Link group
  • Link (knot theory)
  • Milnor conjecture (topology)
  • Milnor map
  • Möbius energy
  • Mutation (knot theory)
  • Physical knot theory
  • Planar algebra
  • Smith conjecture
  • Tait conjectures
  • Temperley–Lieb algebra
  • Thurston–Bennequin number
  • Tricolorability
  • Unknotting number
  • Unknotting problem
  • Volume conjecture

Theorems

  • Schubert's theorem
  • Conway's theorem
  • Alexander's theorem

Lists

  • List of mathematical knots and links
  • List of prime knots