Trefoil Probabilities and Polyhedra

Joseph Geisz; Chris Peterson; Clayton Shonkwiler

Home/Archive/arXiv

arXiv · 2609.24860

Trefoil Probabilities and Polyhedra

Joseph Geisz·Chris Peterson·Clayton Shonkwiler

View original source ↗Read source document ↗Export metadata ↓

Abstract

We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure $dμ(x,y)=\frac{dx dy}{π (1+x^2+y^2)^2}$. Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $μ$-content of a random triangle. This yields a regular octahedral probability of $\frac{15}{4π^2}$ and a trefoil probability of $\frac{1}{16π^2}$.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Joseph Geisz, Chris Peterson, Clayton Shonkwiler. 2026-09-21. Trefoil Probabilities and Polyhedra. https://arxiv.org/abs/2609.24860

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

math.GT↗

arXiv2026

Knots and the Sierpinski Tetrahedron

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.