Chau and Shah (arXiv:2603.28487) call a finite simple graph G TB-symmetrical when, for every two cycle lengths r ≠ s, the numbers of r-cycles through each edge, through each pair of adjacent edges and—with a sign recording orientation—through each pair of n…
Gabrovšek and Cavicchioli have tabulated the 427 prime spherical knotoids with at most seven crossings. Their table divides them into 37 rotatable and 390 non-rotatable knotoids, but the non-rotatable column is marked as conjectural: for 28 of the seven-cro…
A marking of a graph Γ with vertex set V is a map R: V → AutΓ; it makes V into a right quasigroup, and by a theorem of Ta that right quasigroup is a rack exactly when R_(Rᵥ(w))=RᵥR_wRᵥ⁻¹ for all v,w ∈ V. Let μ_(rack)(Γ) and μ_(qnd)(Γ) be the number of marki…
An origami is a pair (h,v) of permutations of n symbols generating a transitive group, taken up to simultaneous conjugation; SL(2,ℤ) acts through T(h,v)=(h,vh⁻¹) and S(h,v)=(hv⁻¹,v), and the stabiliser of an origami is its Veech group. Jeffreys and Matheus…
The non-orientable four-genus γ₄(K) of a knot K ⊂ S³ is the least first Betti number of a smooth, compact, properly embedded surface in B⁴ with boundary K; γ₄(K)=1 says that K bounds a Möbius band. Lee and Sabloff have recently surveyed γ₄ over all knots of…
Framba has recently defined an equivariant grid homology for strongly invertible knots and, from it, two integer invariants τ₀ and τ₁ satisfying τ₀(K) ≤ τ(K) ≤ τ₁(K). That paper computes the two invariants for no knot, and asks for a strongly invertible kno…
For a positive integer N and a unit k of ℤ/N, the linear quandle Λ_(N,k) is the Alexander quandle of ℤ/N with φₖ(x)=kx; it is the dihedral quandle R_N when k=-1 and the trivial quandle when k=1. Ta tabulated the number |GoodΛ_(N,k)| of good involutions of e…
A CNS-quandle is a connected, noninvolutory quandle equipped with a good involution. Ta asked for which integers k a CNS-quandle of order k exists, and answered the question for k ≤ 8, where a single one occurs, the symmetric Galkin quandle of order 6. We s…
A marking of a finite (di)graph Γ with vertex set V is a map R: V → AutΓ; it makes V into a right quasigroup, and by a theorem of Ta that right quasigroup is a rack exactly when R_(Rᵥ(w))=RᵥR_wRᵥ⁻¹ for all v,w. Ta tabulated the numbers μ_(rack)(Γ) and μ_(qn…
