Quandle Homology Groups, Their Betti Numbers, and Virtual Knots

Details Quandle Homology Groups, Their Betti Numbers, and Virtual Knots Quandle Homology Groups, Their Betti Numbers, and Virtual Knots

1999-09-28

arxiv.org/abs/math/9909161v1

Mathematics

Geometric Topology

22 pages, about 6 figures (eps), latex

Scientific paper

Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluationg the state-sum invariants defined from quandle cohomology.

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