Homology of dihedral quandles

Maciej Niebrzydowski, Jozef H. Przytycki

Published 2006-11-27Version 1

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H_3^R(R_p) = Z \oplus Z_p for p odd prime. We also show that H_n^R(R_p) contains Z_p for n>2. Furthermore, we show that the torsion of H_n^R(R_3) is annihilated by 3. We also prove that the quandle homology H_4^Q(R_p) contains Z_p for p odd prime. We conjecture that for n>1 quandle homology satisfies: H_n^Q(R_p) = Z_p^{f_n}, where f_n are "delayed" Fibonacci numbers, that is, f_n = f_{n-1} + f_{n-3} and f(1)=f(2)=0, f(3)=1. Our paper is the first step in approaching this conjecture.