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The second quandle homology of the Takasaki quandle of an odd order abelian group
by
Jozef H. Przytycki
GWU and UTD
Coauthors: Maciej Niebrzydowski (ULL)
We prove that if G is an abelian group of odd order and T(G) its Takasaki quandle (that is a*b=2b-a) then there is an isomorphism from the second quandle homology H2Q(T(G)) to G ∧G where ∧ is the exterior product. In particular, for G=Zkn, k odd we have H2Q(T(Zkn)) = Zkn(n-1/2. Our proof is performed in five steps: First, we construct Cayley graph and Cayley 2-simplex of T(G). Then we choose a spanning tree for the Cayley graph and contract it. The result is the group Z(G×G) divided by relations [x, x]=0=[0, x] and [x, z]+[z, y] = [x, z-y+x]+[z-y+x, y]. Then we prove that for G generated by 2 elements the main result holds, in particular that [x, y] = -[y, x] . In the fourth step we show that the relation [z+x, z+y] = [z, y]+[x, z]+[x, y] holds and finally it leads to 2([w, z+y]-[w, z]-[w, y])=0 which for 2 not zero divisor, leads to linearity on the second component and from skew-symmetry to bilinearity and exterior product. Our result can be directly applied to classical knots as 2-(co)cycles give knot invariants. They can be also used to produce new nontrivial quandles.
Date received: March 26, 2010
Copyright © 2010 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbaf-04.