Homology of dihedral quandles II

Jozef H. Przytycki

Knots in Washington XXIII; Quandles, their homology and ramifications
November 17-19, 2006
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), przytyck@gwu.edu, Yongwu Rong (GWU), rong@gwu.edu, Alexander Shumakovitch (GWU), shurik@gwu.edu

Conference Homepage

Homology of dihedral quandles II
by
Jozef H. Przytycki
George Washington University
Coauthors: Maciej Niebrzydowski (GWU)

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of the dihedral quandle satisfies H₃^R(R_p) = Z ⊕Z_p for p odd prime (Ohtsuki, Conjecture 5.12). In particular, the quandle homology H₃^Q(R_p) = Z_p. We conjecture that for n > 1 the 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. We propose the method of approaching the conjecture by constructing rack homology operations H^R_n(R_p) → H^R_n+1(R_p) and H^R_n(R_p) → H^R_n+2(R_p), and quandle homology operations H^R_n(R_p) → H^R_n+2(R_p) and H^R_n(R_p) → H^R_n+3(R_p). We conjecture, and partially prove (as outlined in the previous talk by Maciej Niebrzydowski), that the above operations are monomorphisms and the images (in appropriate dimensions) are disjoint. To approach the general conjecture about H_n^Q(R_p) = Z_p^f_n we need one more quandle homology operation H^Q_n(R_p) → H^Q_n+4(R_p), construction of which is an open problem.

References: T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geometry and Topology Monographs, Volume 4, 2003, 455-465; e-print: http://front.math.ucdavis.edu/math.GT/0406190

Date received: November 15, 2006