A table of $\theta$-curves and handcuff graphs with up to seven crossings

Abstract

We enumerate all the $\theta$-curves and handcuff graphs with up to seven crossings by using algebraic tangles and prime basic $\theta$-polyhedra. Here, a $\theta$-polyhedron is a connected graph embedded in a 2-sphere, whose two vertices are 3-valent, and the rest are 4-valent. There exist twenty-four prime basic $\theta$-polyhedra with up to seven 4-valent vertices. We can obtain a $\theta$-curve and handcuff graph diagram from a prime basic $\theta$-polyhedron by substituting algebraic tangles for their 4-valent vertices.