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Moriuchi, Hiromasa

An enumeration of theta-curves with up to seven crossings. (English) Zbl 1200.57002

J. Knot Theory Ramifications 18, No. 2, 167-197 (2009).
This article concerns the enumeration of prime \(\theta\)-curves. A \(\theta\)-curve is a graph in the \(3\)-sphere consisting of two vertices and three edges, each edge joining the two vertices. Two \(\theta\)-curves are called equivalent (and ultimately considered the same) if there exists a homeomorphism of the 3-sphere onto itself sending the \(\theta\)-curves into one another. Apparently, Litherland has produced a list of prime \(\theta\)-curves up to seven crossings. The motivation of this article is to prove that this list is complete. For this the author adapts the methods of Conway to assign knots to \(\theta\)-curves and uses the Yamada invariant to distinguish the elements in his list.
Reviewer: Pedro Lopes (Nottingham)

Cited in 12 Documents

MSC:

57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

prime \(\theta\)-curve; prime basic \(\theta\)-polyhedron; Yamada polynomial
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