Combined injectivity conjecture for quandle homology operations on odd dihedral quandles

Let RkR_k be an odd dihedral quandle, and let (hs)(h_s)_* and (h(s,0))(h'_{(s,0)})_* be the induced operations

(hs):Hn+1Q(Rk)Hn+3Q(Rk),(h_s)_*:H_{n+1}^Q(R_k)\longrightarrow H_{n+3}^Q(R_k), (h(s,0)):HnQ(Rk)Hn+3Q(Rk).(h'_{(s,0)})_*:H_n^Q(R_k)\longrightarrow H_{n+3}^Q(R_k).

Here h(s,0)=h0hsh'_{(s,0)}=h'_0h_s. Combined quandle injectivity conjecture. For odd kk and n>1n>1, the map

(hs)(h(s,0)):Hn+1Q(Rk)HnQ(Rk)Hn+3Q(Rk)(h_s)_*\oplus(h'_{(s,0)})_*:H_{n+1}^Q(R_k)\oplus H_n^Q(R_k)\longrightarrow H_{n+3}^Q(R_k)

is a monomorphism. Because the analogous composition involving two degree-one operations induces zero on homology, the source proposes this composition instead; the conjecture remains open.

Sources & referencesView supporting material

Primary source

Maciej Niebrzydowski and Jozef H. Przytycki, “Homology of dihedral quandles”, arXiv:math/0611803 (2006).

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