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4.3 DT (Dowker-Thistlethwaite) Codes

The DT Code (``DT'' after Clifford Hugh Dowker and Morwen Thistlethwaite) of a knot $ K$ is obtained as follows:

Figure 3: A knot with DT code 8, 10, 2, 12, 4, 6
\begin{figure}\centering {
\includegraphics[width=2in]{figs/DTExample.eps}
}
\end{figure}

KnotTheory` has some rudimentary support for DT codes:

In[1]

In[2]:= ?DTCode
DTCode[i1, i2, ...] represents a knot via its DT (Dowker-Thistlethwaite) code. DTCode also acts as a "type caster", so for example, DTCode[K] where K is is a named knot returns the DT code of that knot.

Thus for example, the DT codes for the last 9 crossing alternating knot $ 9_{41}$ and the first 9 crossing non alternating knot $ 9_{42}$ 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]]


next up previous contents index
Next: 4.4 Braid Representatives Up: 4 Presentations Previous: 4.2 Gauss Codes   Contents   Index
Dror Bar-Natan 2005-09-14