List of knot theory topics
Appearance
(Redirected from Outline of knot theory)
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
[edit]Knots, links, braids
[edit]- 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
[edit]- 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
[edit]- 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:
Tangles
[edit]- 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
[edit]Operations
[edit]Elementary treatment using polygonal curves
[edit]- elementary move (R1 move, R2 move, R3 move)
- R-equivalent
- delta-equivalent
Invariants and properties
[edit]- 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
[edit]- 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