Conjecture 5.12 — (R.

Definitions and mathematical statement

For a prime $p$, let the dihedral rack/quandle $R_p$ be $\mathbb Z/p\mathbb Z$ with $x*y=2y-x$. In the rack-homology interpretation supplied by the summary, take $C_n^R(R_p)=\mathbb Z[R_p^n]$ and boundary \[ \partial_n(x_1,\ldots,x_n)=\sum_{i=2}^n(-1)^i \left[(x_1,\ldots,\widehat{x_i},\ldots,x_n) -(x_1*x_i,\ldots,x_{i-1}*x_i,x_{i+1},\ldots,x_n)\right]. \] The degree-three group is $H_3^R=\ker\partial_3/\operatorname{im}\partial_4$, with integer coefficients.

The extracted conjecture says \[ H_3(R_p)\cong\mathbb Z\oplus\mathbb Z/p\mathbb Z \quad\text{for }p\text{ prime}. \] Its homology superscript is missing, while the summary identifies rack homology and reports the formula proved for odd primes. The prime $2$ cannot be silently included under the displayed rack convention: $R_2$ has trivial operation, all boundaries vanish, and $H_3^R(R_2)\cong\mathbb Z^8$. Thus the all-prime wording needs correction or a different intended convention; the coherent reported theorem is the odd-prime rack statement.

The summary attributes that odd-prime result to Niebrzydowski--Przytycki, using Mochizuki's mod-$p$ quandle computation and a rack/quandle homology decomposition. It also reports $p$-torsion in higher rack degrees $n\ge3$ and related $R_3$ results, without exact subsidiary hypotheses. Those auxiliary reports are not substituted for the degree-three target or independently certified. The registry solved label is retained as a literature attribution, with the exceptional-prime issue explicit.

Short English statement

Compute the third integer rack homology of the dihedral quandle. The supplied literature report proves $\mathbb Z\oplus\mathbb Z/p$ for odd primes; the extracted all-prime wording fails at $p=2$ under the standard rack definition.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400108$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 5.12, attributed to R. Fenn, C. Rourke and B. Sanderson, PDF page 93: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29.
• M. Niebrzydowski and J. H. Przytycki, "Homology of dihedral quandles", Journal of Pure and Applied Algebra 213 (2009), 742--755; arXiv:math/0611803 (2006), as supplied by the summary.
• Mochizuki's computation $H_3^Q(R_p;\mathbb Z_p)\cong\mathbb Z_p$ and a rack/quandle splitting are cited by the summary without separate full references.