Naming and Enumeration: Difference between revisions
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<tt>Knot[n, k] denotes the kth knot with n crossings in the Rolfsen table. Knot[11, Alternating, k] denotes the kth alternating 11-crossing knot in the Hoste-Thistlethwaite table. Knot[11, NonAlternating, k] denotes the kth non alternating 11-crossing knot in the Hoste-Thistlethwaite table.</tt> |
<tt>Knot[n, k] denotes the kth knot with n crossings in the Rolfsen table. Knot[11, Alternating, k] denotes the kth alternating 11-crossing knot in the Hoste-Thistlethwaite table. Knot[11, NonAlternating, k] denotes the kth non alternating 11-crossing knot in the Hoste-Thistlethwaite table.</tt> |
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<tt>Link[n, Alternating, k] denotes the kth alternating n-crossing link in the Thistlethwaite table. Link[n, NonAlternating, k] denotes the kth non alternating n-crossing link in the Thistlethwaite table.</tt> |
<tt>Link[n, Alternating, k] denotes the kth alternating n-crossing link in the Thistlethwaite table. Link[n, NonAlternating, k] denotes the kth non alternating n-crossing link in the Thistlethwaite table.</tt> |
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Thus, for example, let us verify that the knots [[6_1]] and [[9_46]] have the same Alexander polynomial: |
Thus, for example, let us verify that the knots [[6_1]] and [[9_46]] have the same Alexander polynomial: |
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<!--$$Alexander[Knot[6, 1]][t]$$--> |
<!--$$Alexander[Knot[6, 1]][t]$$--> |
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<tt>In[4]:=</tt><code> Alexander[Knot[6, 1]][t]</code> |
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<tt>Out[4]=</tt> <math>-2 t+5-\frac{2}{t}</math> |
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<!--$$Alexander[Knot[9, 46]][t]$$--> |
<!--$$Alexander[Knot[9, 46]][t]$$--> |
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<tt>In[5]:=</tt><code> Alexander[Knot[9, 46]][t]</code> |
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<tt>Out[5]=</tt> <math>-2 t+5-\frac{2}{t}</math> |
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Revision as of 15:03, 23 August 2005
KnotTheory` comes loaded with some knot tables; currently, the Rolfsen table of prime knots with up to 10 crossings [Rolfsen], the Hoste-Thistlethwaite tables of prime knots with up to 16 crossings and the Thistlethwaite table of prime links with up to 11 crossings (see Further Knot Theory Software#Knotscape):
(For In[1] see Setup)
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In[2]:= Knot[n, k] denotes the kth knot with n crossings in the Rolfsen table. Knot[11, Alternating, k] denotes the kth alternating 11-crossing knot in the Hoste-Thistlethwaite table. Knot[11, NonAlternating, k] denotes the kth non alternating 11-crossing knot in the Hoste-Thistlethwaite table. |
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In[3]:= Link[n, Alternating, k] denotes the kth alternating n-crossing link in the Thistlethwaite table. Link[n, NonAlternating, k] denotes the kth non alternating n-crossing link in the Thistlethwaite table. |
Thus, for example, let us verify that the knots 6_1 and 9_46 have the same Alexander polynomial:
In[4]:= Alexander[Knot[6, 1]][t]
Out[4]=
In[5]:= Alexander[Knot[9, 46]][t]
Out[5]=
References
[Rolfsen] ^ D. Rolfsen, Knots and Links, Publish or Perish, Mathematics Lecture Series 7, Wilmington 1976.
