A classifying invariant of knots, the knot quandle

  title={A classifying invariant of knots, the knot quandle},
  author={David Joyce},
  journal={Journal of Pure and Applied Algebra},
  • D. Joyce
  • Published 1982
  • Mathematics
  • Journal of Pure and Applied Algebra
Abstract The two operations of conjugation in a group, x▷y=y -1 xy and x▷ -1 y=yxy -1 satisfy certain identities. A set with two operations satisfying these identities is called a quandle. The Wirtinger presentation of the knot group involves only relations of the form y -1 xy = z and so may be construed as presenting a quandle rather than a group. This quandle, called the knot quandle, is not only an invariant of the knot, but in fact a classifying invariant of the knot. 
To be or knot to be
Abstract A quandle is a set with two operations that satisfy three conditions. For example, there is a quandle naturally associated to any group. It turns out that one can associate a quandle to any
Inequivalent surface-knots with the same knot quandle
Quandles at Finite Temperatures II
The number of colorings of a knot diagram by a given quandle is a knot invariant. This follows naturally from the fact that the knot quandle is a knot invariant, and is observed again in the context
Knot quandle decomposition along a torus
We study the structure of the augmented fundamental quandle of a knot whose complement contains an incompressible torus. We obtain the relationship between the fundamental quandle of a satellite knot
A multiple conjugation quandle and handlebody-knots
Abstract We introduce a notion of multiple conjugation quandles and give some examples of it. A multiple conjugation quandle has a mixed structure of a group and a quandle. We see that the axioms of
Quandles at Finite Temperatures I
The number of colorings of a knot diagram by a quandle has been shown to be a knot invariant by CJKLS using quandle cohomology methods. In a previous paper by the second named author, the CJKLS
Polynomial quandle cocycles, their knot invariants and applications
A quandle is a set with a binary operation that satisfies three axioms that correspond to the three Reidemeister moves on knot diagrams. Homology and cohomology theories of quandles were introduced
Quandle cohomology and state-sum invariants of knotted curves and surfaces
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation
On colorability of knots by rotations, Torus knot and PL trochoid
The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on
A note on the sheet numbers of twist-spun knots
The sheet number of a 2-knot is a quantity which reflects the complexity of the knotting in 4-space. The aim of this note is to determine the sheet numbers of the 2and 3-twist-spun trefoils. For this


Characterizations of knots and links
Although there are inequivalent classical knots with isomorphic groups, J. Simon recently characterized each knot type by a group: the free product of two, suitably chosen, cable-knot groups [4]. In
A Group to Classify Knots
Simon [2] has shown that we can effectively associate to each (tame) knot K in S a finitely presented classifying group CGR(K) so that two knots K and K' are equivalent if and only if CGR(K) and
Abstractions of symmetric functions
  • Tohoku Math. J
  • 1943
Abstractions of symmetric functions
  • Tohoku Math. J
  • 1943