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 together 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 ; 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 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 torus knot.
- * Figure-eight knot the only 4-crossing knot
- * Granny knot and Square knot 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, 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
- Conway notation
- Dowker notation
General knot types
- 2-bridge knot
- Alternating knot; a knot that can be represented by an alternating diagram.
- 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.
- 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
- 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
- Algebraic link
- Hyperbolic link
- Pretzel link
- Split link
- String link
Tangles
- Tangle
- Algebraic tangle
Braids
- Braid theory
- * Braid group
Operations
- Band sum
- Flype
- Fox n-coloring
- * Tricolorability
- Knot sum
- Reidemeister move
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. 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
- Eilenberg-Mazur swindle
- Fary-Milnor theorem
- Gordon–Luecke theorem
- Khovanov homology
- Knot group
- Knot tabulation
- Knot theory
- Knotless embedding
- Linkless embedding
- Link concordance
- Link group
- Link
- Milnor conjecture
- Milnor map
- Möbius energy
- Mutation
- Physical knot theory
- Planar algebra
- Smith conjecture
- Tait conjectures
- Temperley–Lieb algebra
- Thurston–Bennequin number
- Tricolorability
- Unknotting number
- Unknotting problem
- Volume conjecture
Lists
- List of mathematical knots and links
- * List of prime knots