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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.

History

Knots, links, braids

• 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, 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 6
  • Three-twist knot is the twist knot with three-half twists, also known as the 5₂ knot.
  • Trefoil knot A knot with crossing number 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.
• Knots and graphs general introduction to knots with mention of Reidemeister moves

Notation used in knot theory:

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

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 tangent to the standard contact structure.
• Lissajous knot
• Ribbon knot
• Satellite knot
• Slice knot
• Torus knot
• Transverse knot
• Twist knot
• Virtual knot
• welded knot
• Wild knot

Links

• 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^1/2 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–Wenzl algebra
• Clasper (mathematics)
• Eilenberg–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