Sometimes it is useful to work with weighted graphs of KLs given by a list of unordered pairs and list of vertex signs instead of Gauss or Dowker codes. The LinKnot function fGraphInc (webMathematica fGraphInc) calculates from a Conway symbol, Dowker code, or P-data of any KL the corresponding graph of that KL given by edges (as a list of unordered pairs) and by the list of vertex signs.
The LinKnot function fPlanarEmbKL (webMathematica fPlanarEmbKL) calculates the planar embedding of a prime KL given by a Conway symbol, P-data or Dowker code. An output is the list that consists of the graph of the input KL, its planar embedding given by vertex cycles, and the faces of the planar embedded graph. The basis of this program is the external program planarity.exe written by J.M. Boyer (Boyer and Myrvold, 2004).
As we already mentioned, every KL shadow is a 4-valent graph. If we have any polyhedral graph G, we can obtain its corresponding mid-edge graph M(G) defined by mid-edge points of G by connecting mid-edge points belonging to adjacent edges of G. Clearly, the result M(G) is always a 4-valent graph. For example, for the tetrahedron graph
. The LinKnot function fMidEdgeGraph (webMathematica fMidEdgeGraph) gives a mid-edge graph for any polyhedral graph G given by a list of unordered pairs of vertices. Every 4-regular graph represents a shadow of a KL. From it, we can find its corresponding alternating KL diagram given by Dowker code. The function fKLfromGraph (webMathematica fKLfromGraph) gives the Dowker code in the DT-form (in Knotscape format) of a KL defined by a given 4-regular graph G. The corresponding Dowker code with signs can be obtained from it by using the function fSignsKL (webMathematica fSignsKL).
From a signed graph of KL we can recover KL from which it originated by constructing mid-edge graph, where to every digon in the KL graph corresponds a digon in the mid-edge graph.