Virtual-knot problem 22 — The Rack Space

Definitions and mathematical statement

A rack is a set $X$ with an operation $x\triangleright y$ such that each right translation is bijective and \[ (x\triangleright y)\triangleright z =(x\triangleright z)\triangleright(y\triangleright z). \] In the standard rack-space convention, the chain group is $C_n^R(X)={\mathbb{Z}}[X^n]$, with boundary \[ \partial(x_1,\ldots,x_n)=\sum_{i=1}^n(-1)^i \left[(x_1,\ldots,\widehat{x_i},\ldots,x_n) -(x_1\triangleright x_i,\ldots,x_{i-1}\triangleright x_i, x_{i+1},\ldots,x_n)\right]. \] Its homology is the integral homology of the usual rack space. The problem is to develop the rack-space homology program and, in particular, investigate the properties and associated homology of the birack space. A birack involves an invertible nondegenerate set-theoretic Yang--Baxter crossing operation; the exact birack axioms and corresponding space construction are not specified in this excerpt, so no particular birack chain model is chosen.

The source reports geometric interpretations: two-dimensional cycles are represented by consistently rack-coloured virtual link diagrams, and three-dimensional cycles by such diagrams with regions also coloured, referring to Greene's thesis. The required crossing/region colour rules and framing conventions are omitted. It says second homology of the dihedral rack is treated in that thesis, and that for prime $p$, third homology has a factor ${\mathbb{Z}}_p$. The usual dihedral rack has underlying set ${\mathbb{Z}}/p{\mathbb{Z}}$ and operation $x\triangleright y=2y-x$. The source neither defines factor nor states whether $p$ must be odd; the prime $2$ requires checking, since that operation becomes trivial there. No missing second-homology formula or direct-summand assertion is invented.

Short English statement

Investigate rack-space homology through coloured virtual-link diagrams, and develop the corresponding birack-space and homology theory, retaining the stated but incompletely specified dihedral-rack results as background.

Sources

• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory, revised survey, problem-list item 22, "The Rack Space." https://arxiv.org/abs/1409.2823. Recorded access 2026-07-29; extraction source-tex.
• The statement credits rack spaces to Fenn, Rourke and Sanderson, names further homology work of Carter, Kamada and Saito, and refers to Greene's thesis for coloured-cycle interpretations and second dihedral-rack homology. All these citation entries and the reference for the ${\mathbb{Z}}_p$ factor are missing from the extraction.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10600022$, exact source code catalogue entry; supplied partially_solved/PARTIAL-PROGRESS status. Its summary reports the birack question unresolved without a newer precise citation.