List of knot theory topics

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.

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 $$\mathbb R^3$$ 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 t1/2 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