Naming and Enumeration

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)

In[2]:= ?Knot 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.

In[3]:= ?Link 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]= $-2t+5-{\frac {2}{t}}$

In[5]:= Alexander[Knot[9, 46]][t]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 t+5-\frac{2}{t}}

We can also check that the Borromean rings, L6a4 in the Thistlethwaite table, is a 3-component link:

Thus at the moment there are 802 knots and 1424 links known to KnotTheory`:

References

[Rolfsen] ^  D. Rolfsen, Knots and Links, Publish or Perish, Mathematics Lecture Series 7, Wilmington 1976.