Bridges 2011 Regular Paper

This paper contains a survey of different methods for generating knots and links based on geometric polyhedra, suitable for applications in chemistry, biology, architecture, sculpture (or jewelry). We describe several ways of obtaining 4-valent polyhedral graphs and their corresponding knots and links from geometrical polyhedra: mid-edge construction, cross-curve and double-line covering, and edge doubling constructions. These methods are implemented in LinKnot and can be applied to the data bases of polyhedra. In a similar way, an edge doubling construction transforms fullerene graphs into alternating knot and link diagrams.