Using Knot Theory to Identify and Enumerate Knots of Circular DNA Molecules

Mathematical and computational tools can help identify the topology of circular DNA molecules when only limited experimental data are available. Knotted DNA occurs as a result of many biological processes. We have created computational tools to determine all knot types corresponding to a fixed regular projection of a knot. A projection is regular if every node in the projection maps back to exactly two points in 3-dimensions. This corresponds for example to the image of a DNA knot obtained under a microscope. In the context of DNA topology we call such image a knot shadow. We are able to identity all possible knot types that can result from the given shadow. In this poster we ask the question: how many different shadows exist of a given crossing number? We thus attempt to enumerate all knot shadows of small degree, or equivalently, to enumerate all 4-valent planar graphs with a small number of nodes. We identify families of knot projections with common properties.