The Betti numbers of some finite racks
By R. Litherland, Sam Nelson. 2001
91 citations recorded by the source corpus.
Abstract We show that the lower bounds for Betti numbers given in (J. Pure Appl. Algebra 157 (2001) 135) are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e10ac3c1da1baa2cbce11bbe80d2920534300455
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Citations in the corpus
- The rack space (2003, background): “Rack cohomology is the cohomology of the rack space and quandle cohomology is a quotient, see Litherland and Nelson [23].”
- Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, background): “2 then yields a decomposition of the corresponding chain complexes into the degenerate and the normalized parts, generalizing the homology decomposition for quandles from [29].”
- Knot invariants derived from quandles and racks (2002, background): “Further calculations on homology/cohomology groups of finite Alexander quandles are found in their papers [21] and [24].”
- Problems on invariants of knots and 3-manifolds (2004, background): “From a table in [264] we omit one of two dual quandles and quandles that are not connected (see remarks on Problem 5.”
- Homology Groups of Symmetric Quandles and Cocycle Invariants of Links and Surface-links (2009, background): “For details on quandle homologies and invariants, refer to [1, 3, 4, 5, 10, 11, 16, 20, 21].”