Homology of dihedral quandles

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Abstract

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H3R(Rp)=ZZp for p odd prime [T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002) 377-572, Conjecture 5.12]. We also show that HnR(Rp) contains Zp for n3. Furthermore, we show that the torsion of HnR(R3) is annihilated by 3. We also prove that the quandle homology H4Q(Rp) contains Zp for p odd prime. We conjecture that for n>1 quandle homology satisfies: HnQ(Rp)=Zpfn, where fn are “delayed” Fibonacci numbers, that is, fn=fn1+fn3 and f(1)=f(2)=0,f(3)=1. Our paper is the first step in approaching this conjecture.

MSC

55N35
18G60
57M25

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