Extensions of Quandles and Cocycle Knot Invariants

By J. Scott Carter, Mohamed Elhamdadi, Marina Appiou Nikiforou, Masahico Saito. 2001

76 citations recorded by the source corpus.

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26c93055a1be3d45c1638b2f5af57c0e60caada7

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In Artificial Intelligence

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Citations in the corpus

Automorphism Groups of Quandles (2010, background): “(70) matrix columns in disjcyc notation: [[[2, 3], [4, 5]], [[1, 5], [3, 6]], [[1, 4], [2, 6]], [[1, 5], [3, 6]], [[1, 4], [2, 6]], [[2, 3], [4, 5]]] (70) inner automorphism group order = 6, not abelian, not cyclic, order seq = {[1, 1], [2, 3], [3, 2]} (70) automorphism group order = 12, not abelian”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “Inspired by the theory of abelian extensions of quandles [4, 13], we define extensions of cycle sets by abelian groups, of which the structure from Lemma 9.”
Quandle coverings and their Galois correspondence (2006, background): “This correspondence has to be generalized in two directions in order to apply to our general setting: • The usual formulation is most appealing for abelian groups Λ, and has been independently developed in [6] and [9].”
Problems on invariants of knots and 3-manifolds (2004, methodology): “Remark It is known, see [78], that each α ∈ H2 Q(X;A) gives an abelian extension A → Y p → X , where Y = A × X , which forms a quandle with the binary operation given by (a1, x1) ∗ (a2, x2) = ( a1 + φ(x1, x2), x1 ∗ x2 ) using a 2-cocycle φ representing α, and p denotes the natural projection.”
Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles (2002, background): “#! +f " 0 1 #! −f " t 1 #! −f " −t+1 0 #! = (δf) " 0 1 # , " 1 0 #! , which is a contradiction. We consider another family of examples that are similar to those considered in [3]. Proposition 2.15 For positive integers q,h,k, let X= Ω(h,k) q = Zq[s±1,t±1]/{(1 − s)h,(1 − t)k}”
Cocycle knot invariants from quandle modules and generalized quandle homology (2003, result): “The fact that~ does not change by Reidemeister moves for fixed base points is similar to the proof in [10] for the knot case and tha t in [6] for the link case, proved for abelian cocycle invariants, except one jus t ob erves that the order of group elements is also preserved under Reidemeister move”