Next: 4.4 Braid Representatives
Up: 4 Presentations
Previous: 4.2 Gauss Codes
  Contents
  Index
The DT Code (``DT'' after
Clifford
Hugh Dowker and Morwen
Thistlethwaite) of a knot is obtained as follows:
KnotTheory` has some rudimentary support for DT codes:
In[2]:= ?DTCode
|
Thus for example, the DT codes for the last 9 crossing alternating knot
and the first 9 crossing non
alternating knot
are:
In[3]:= | dts = DTCode /@ {Knot[9, 41], Knot[9, 42]} |
Out[3]= | {DTCode[6, 10, 14, 12, 16, 2, 18, 4, 8], > DTCode[4, 8, 10, -14, 2, -16, -18, -6, -12]} |
(The DT code of an alternating knot is always a sequence of positive numbers but the DT code of a non alternating knot contains both signs.)
DT codes and Gauss codes carry the same information and are easily convertible:
In[4]:= | gcs = GaussCode /@ dts |
Out[4]= | {GaussCode[1, -6, 2, -8, 3, -1, 4, -9, 5, -2, 6, -4, 7, -3, 8, -5, 9, -7], > GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 9, -7, 4, -8, 6, -9, 7]} |
In[5]:= | DTCode /@ gcs |
Out[5]= | {DTCode[6, 10, 14, 12, 16, 2, 18, 4, 8], > DTCode[4, 8, 10, -14, 2, -16, -18, -6, -12]} |
Conversion between DT codes and/or Gauss codes and PD codes is more complicated; the harder side, going from DT/Gauss to PD, was written by Siddarth Sankaran at the University of Toronto:
In[6]:= | PD[DTCode[4, 6, 2]] |
Out[6]= | PD[X[4, 2, 5, 1], X[6, 4, 1, 3], X[2, 6, 3, 5]] |