PD Notation


A PD notation for an oriented link may be obtained from a diagram as follows as follows. First label its edges from 1 to n as follows, where n is the number of edges. Choose a first component and a starting edge, labeled 1. Label subsequent edges of the component according to its orientation. When this is done, choose a second component and an edge to label k+1 (if the first component had k edges). Continue until all the edges have been labeled.

Secondly, choose a component (not necessarily the first as chosen above) and choose a crossing for which it appears as an undercrossing. Assign an ordered 4-tuple (denoted {a,b,c,d} here) to this crossing by listing the valent edges of the crossing in counterclockwise order, beginning with the incoming lower edge. Repeat for subsequent undercrossings of this component (according to its orientation) to obtain an ordered list of m 4-tuples, where m is the number of times it is an undercrossing. Now repeat for the subsequent components (according to an arbitrary order which may differ from the one above).

Example

We apply the above method to assign a PD notation to the oriented link L4a1{0} (see the figure below). We label the edges from 1 to 8 by choosing the blue component to be the first and by choosing the starting points on the first and second components at 1 and 5 as shown, respectively. Now a crossing is chosen where the purple undercrosses; it is labeled {6,1,7,2} according to the convention above. The next such crossing is {8,3,5,4}. Then, as the blue component undercrosses at the crossings {2,5,3,6}, {4,7,1,8}, the PD notation we obtain is {{6,1,7,2}, {8,3,5,4}, {2,5,3,6}, {4,7,1,8}}. Note that the choices made in this example were tailor-made to agree with the PD notation for L4a1{0} as it appears here.

Remarks

PD Notation (Math)

This just writes the PD Notation for an oriented link in the form the computer program KnotTheory operates with. For example, the PD notation for L4a1{0}, {{6,1,7,2},{8,3,5,4},{2,5,3,6},{4,7,1,8}} , becomes: PD[X[6,1,7,2],X[8,3,5,4],X[2,5,3,6],X[4,7,1,8]].