Enumeration of knotted 4-regular graphs

Knotted 4-regular graphs appear in quantum gravity models as possible states of geometry, since these arise as graphs dual to certain triangulations of three-dimensional spaces. In this paper, methods to enumerate these graphs are discussed. We develop a new graph invariant, which is used to obtain a census of knotted 4-regular graphs represented by Dowker-Thistlethwaite sequences of five nodes or less, with graphs and their mirror images identified in the census. This invariant is used to place a lower limit on the minimal number of nodes for any diagram of a graph.