6 problems

A rack is a set equipped with a binary operation satisfying the rack axioms; two racks are identified when they are isomorphic. Let $n$ denote the size of the rack. Rack enumeratio…

Let $\mathcal{O}$ be a finite rack of type C, let $\mathfrak{q}$ be a faithful $2$-cocycle on $\mathcal{O}$, and let $\mathscr{B}(\mathcal{O},\mathfrak{q})$ denote the associated N…

Fenn–Rourke's conjecture. There should exist a countable sequence of finite racks $(R_i)_{i\geq 1}$ such that the sequence

A finite rack $X$ has profile $1^{m_0}l_1^{m_1}l_2^{m_2}\cdots l_k^{m_k}$ when its inner automorphism has $m_0$ cycles of length $1$, $m_1$ cycles of length $l_1$, and so on. Hayas…

Enumeration conjecture.

Let $\mathbf{BB}_n$ be the blocked-braid group on $n$ strands. A blocked torsion element is known to have order either $2$ or $4$. Order-four conjecture. Every blocked torsion elem…