GEOMETRIC INTERPRETATIONS OF QUANDLE HOMOLOGY

By J. Scott Carter, S. Kamada, Masahico Saito. 2000

120 citations recorded by the source corpus.

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c885adee653592ced1ae7797ff8447da916c405

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In Geometry and Topology

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Citations in the corpus

Quandle coverings and their Galois correspondence (2006, background): “For example, given a diagram of a knot K ⊂ R3, it is comparatively easy to read off a fundamental homology class [K] ∈ H2(QK) and to verify that it is an invariant of the knot [7].”
The Betti numbers of some finite racks (2001, background): “In [CJKS1] it is proved (in Proposition 3.9) that the connecting homomorphism HQ n (X) → HD n−1(X) is the zero map when n = 3, and conjectured that this is so for all n; in [ CJKS3 ] (Theorem 8.2) the case n = 4 is proved.”
Knot invariants derived from quandles and racks (2002, background): “Refer to [13, 14, 15, 17] for some calculations and applications of the rack homology groups, and [4, 5, 6, 7, 21, 24] for the quandle homology groups.”
Extensions of Quandles and Cocycle Knot Invariants (2001, methodology): “d, knot diagrams colored by quandles can be used to study quandle homology groups. This view point was developed in [11, 12, 15] for rack homology and homotopy, and generalized to quandle homology in [6]. In [7], constructions of extensions of quandles using cocycles are given, which are similar to ”
The 2-twist-spun trefoil has the triple point number four (2003, background): “Further studies are found in [2, 4, 5, 6, 7, 15, 21, 27, 31] and [33], for example.”
Moves and invariants for knotted handlebodies (2008, background): “The second invariant is a generalization of the quandle cocycle invariant studied by Carter et al [1; 2] and Satoh [16].”
Virtual knot theory-unsolved problems (2005, background): “Study virtual quandle homology in analogy to quandle homology [10, 33].”
Twisted quandle homology theory and cocycle knot invariants (2001, background): “AMS Classification 57N27, 57N99; 57M25, 57Q45, 57T99 Keywords Quandle homology, cohomology extensions, dihedral quandles, Alexander numberings, cocycle knot invariants”