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Examples of different knots including the trivial knot (top left) and the trefoil knot (below it) A knot diagram of the trefoil knot, the simplest non-trivial knot. In topology, knot theory is the study of mathematical knots.
Lickorish and Kenneth Millett won the 1991 Chauvenet Prize for their paper "The New Polynomial Invariants of Knots and Links". [3] Lickorish was included in the 2019 class of fellows of the American Mathematical Society "for contributions to knot theory and low-dimensional topology". [4]
Three-twist knot is the twist knot with three-half twists, also known as the 5 2 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. Notation used in knot theory: Conway notation
A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e. diagrams representing the same knot have the same polynomial. The converse ...
A polygonal knot is a knot whose image in R 3 is the union of a finite set of line segments. [6] A tame knot is any knot equivalent to a polygonal knot. [6] [Note 2] Knots which are not tame are called wild, [7] and can have pathological behavior. [7] In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for ...
In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines. Equivalently, the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot.
Knot theory is a branch of topology that concerns itself with abstract properties of mathematical knots — the spatial arrangements that in principle could be assumed by a closed loop of string. The main article for this category is Knot theory .
A reduced diagram is one in which all the isthmi are removed. Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots.