Naming and Enumeration

2008-07-13T20:55:30Z

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eroloroba
{{Manual TOC Sidebar}}

<code>KnotTheory`</code> comes loaded with some knot tables; currently, the Rolfsen table of prime knots with up to 10 crossings {{ref|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|Knotscape]]):

{{Startup Note}}



{{HelpLine|
n = 2 |
in = <nowiki>Knot</nowiki> |
out= <nowiki>Knot[n, k] denotes the kth knot with n crossings in the Rolfsen table. Knot[n, Alternating, k] (for n between 11 and 16) denotes the kth alternating n-crossing knot in the Hoste-Thistlethwaite table. Knot[n, NonAlternating, k] denotes the kth non alternating n-crossing knot in the Hoste-Thistlethwaite table.</nowiki>}}




{{HelpLine|
n = 3 |
in = <nowiki>Link</nowiki> |
out= <nowiki>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.</nowiki>}}


{{Knot Image Pair|6_1|gif|9_46|gif}}

Thus, for example, let us verify that the knots [[6_1]] and [[9_46]] have the same Alexander polynomial:



{{InOut|
n = 4 |
in = <nowiki>Alexander[Knot[6, 1]][t]</nowiki> |
out= <nowiki> 2
5 - - - 2 t
t</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Alexander[Knot[9, 46]][t]</nowiki> |
out= <nowiki> 2
5 - - - 2 t
t</nowiki>}}


{{Knot Image|L6a4|gif}}

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



{{InOut|
n = 6 |
in = <nowiki>Length[Skeleton[Link[6, Alternating, 4]]]</nowiki> |
out= <nowiki>3</nowiki>}}




{{HelpLine|
n = 7 |
in = <nowiki>AllKnots</nowiki> |
out= <nowiki>AllKnots[] return a list of all knots with up to 11 crossings. AllKnots[n_] returns a list of all knots with n crossings, up to 16. AllKnots[{n_,m_}] returns a list of all knots with between n and m crossings, and AllKnots[n_,Alternating|NonAlternating] returns all knots with n crossings of the specified type.</nowiki>}}




{{HelpLine|
n = 8 |
in = <nowiki>AllLinks</nowiki> |
out= <nowiki>AllLinks[] return a list of all links with up to 11 crossings. AllLinks[n_] returns a list of all links with n crossings, up to 12.</nowiki>}}


Thus at the moment there are 1701936 knots and 5700 links known to <code>KnotTheory`</code>:



{{InOut|
n = 9 |
in = <nowiki>Length /@ {AllKnots[{0,16}], AllLinks[{2,12}]}</nowiki> |
out= <nowiki>{1701936, 5700}</nowiki>}}




{{Graphics|
n = 10 |
in = <nowiki>Show[DrawPD[Knot[13, NonAlternating, 5016], {Gap -> 0.025}]]</nowiki> |
img= Naming_and_Enumeration_Out_10.gif |
out= <nowiki>-Graphics-</nowiki>}}


(Shumakovitch had noticed that this nice knot has interesting Khovanov homology; see {{ref|Shumakovitch}}).

{{Knot Image|T(5,3)|jpg}}

In addition to the tables, KnotTheory` also knows about torus knots:



{{HelpLine|
n = 11 |
in = <nowiki>TorusKnot</nowiki> |
out= <nowiki>TorusKnot[m, n] represents the (m,n) torus knot.</nowiki>}}



{{HelpLine|
n = 12 |
in = <nowiki>TorusKnots</nowiki> |
out= <nowiki>TorusKnots[n_] returns a list of all torus knots with up to n crossings.</nowiki>}}


For example, the torus knots [[T(5,3)]] and T(3,5) have different presentations with different numbers of crossings, but they are in fact isotopic, and hence they have the same invariants (and in particular the same type 3 Vassiliev invariant <math>V_3</math>):



{{InOut|
n = 13 |
in = <nowiki>Crossings /@ {TorusKnot[5, 3], TorusKnot[3, 5]}</nowiki> |
out= <nowiki>{10, 12}</nowiki>}}




{{InOut|
n = 14 |
in = <nowiki>Vassiliev[3] /@ {TorusKnot[5, 3], TorusKnot[3, 5]}</nowiki> |
out= <nowiki>{20, 20}</nowiki>}}


KnotTheory` knows how to plot torus knots; see [[Drawing with TubePlot]].

You can also use the function Knot to parse certain string representations of named knots:



{{InOut|
n = 15 |
in = <nowiki>Knot /@ {"K11a14", "11a_14", "L8a1", "T(3,5)"}</nowiki> |
out= <nowiki>{Knot[11, Alternating, 14], Knot[11, Alternating, 14],

Link[8, Alternating, 1], TorusKnot[3, 5]}</nowiki>}}


In the opposite direction, the function NameString produces the standard name for a knot, used throughout the Knot Atlas.



{{InOut|
n = 16 |
in = <nowiki>NameString /@ {Knot[11, Alternating, 14], TorusKnot[3,5]}</nowiki> |
out= <nowiki>{K11a14, T(3,5)}</nowiki>}}


==References==

{{note|Rolfsen}} D. Rolfsen, ''Knots and Links'', Publish or Perish, Mathematics Lecture Series 7, Wilmington 1976.

{{note|Shumakovitch}} A. Shumakovitch, ''Torsion of the Khovanov Homology'', [http://front.math.ucdavis.edu/math.GT/0405474 arXiv:math.GT/0405474].

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Naming and Enumeration