Notation#

Description of two key diagram encodings that KnotPy supports natively for knots, links, and spatial graphs: - EM code (Ewing–Millett code) [1] - PD code (Planar Diagram code) [2]

Both encodings are supported by KnotPy, though the internal representation is structured most closely to EM codes.—

Definitions and Examples#

### EM Code

Ewing–Millett (EM) encoding — informal specification#

  1. Crossings. Enumerate every crossing by any hashable, totally ordered key (commonly letters a, b, c, …, but any comparable objects are fine).

  2. Half-arcs. At each crossing x, index the four incident half-arcs by integers 0, 1, 2, 3 in counterclockwise order, with the undercrossing half-arcs assigned the even indices (0 and 2).

  3. Endpoints. An endpoint is a pair (x, i) where x is a crossing and i ∈ {0,1,2,3} is a half-arc index. Thus crossing a is incident to (a, 0), (a, 1), (a, 2), (a, 3).

  4. Local connection data (the EM record). For each crossing x, record an ordered 4-tuple of endpoints giving, in order, where (x, 0), (x, 1), (x, 2), (x, 3) continue to (i.e., which endpoints they connect to along the same arc).

    Example mapping (your data, normalized):

    • a → ((b, 3), (b, 2), (c, 3), (c, 2))

    • b → ((d, 3), (e, 0), (a, 1), (a, 0))

    • c → ((e, 3), (d, 0), (a, 3), (a, 2))

    • d → ((c, 1), (e, 2), (e, 1), (b, 0))

    • e → ((b, 1), (d, 2), (d, 1), (c, 0))

  5. Arcs. An (unoriented) arc is the unordered pair of endpoints that are connected. For the example above, the arcs are:

    {(a, 0), (b, 3)}, {(a, 1), (b, 2)}, {(a, 2), (c, 3)}, {(a, 3), (c, 2)}, {(b, 0), (d, 3)}, {(b, 1), (e, 0)}, {(c, 0), (e, 3)}, {(c, 1), (d, 0)}, {(d, 1), (e, 2)}, {(d, 2), (e, 1)}

References#