Printable Manual: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
No edit summary
Line 1: Line 1:
<!-- don't edit independently of [[Manual TOC]]. See discussion page. -->
<!-- don't edit! This page is automatically generated from [[Manual TOC]]. See discussion page. -->


Return to the wiki [[The Mathematica Package KnotTheory`|manual]].
Return to the wiki [[The Mathematica Package KnotTheory`|manual]].

Revision as of 10:40, 29 August 2005


Return to the wiki manual.

Acknowledgement

This Atlas is partially (and indirectly) supported by NSERC grant RGPIN 262178. As a Wiki project, it doesn't make sense anymore to acknowledge individual contributors. Yet for historical purposes, here's our acknowledgement as of the conversion to Wiki formnat on August 2005:

Setup

Start by downloading the file KnotTheory.zip (around 15MB), and unpack it. This will create a subdirectory KnotTheory/ in your current working directory. This done, no installation is required (though you may wish to check out Further Data Files and/or Setting the Path below). Start Mathematica and you're ready to go:

In[2]:= << KnotTheory`

Loading KnotTheory` version of March 22, 2011, 21:10:4.67737. Read more at http://katlas.org/wiki/KnotTheory.

Notice the little "prime" at the end of KnotTheory above. It is a backquote (find it on the upper left side of most keyboards) and not a quote, and it really has to be there for things to work.

Let us check that everything is working well:

In[3]:= Alexander[Knot[6, 2]][t]
Out[3]= -2 3 2 -3 - t + - + 3 t - t t
In[4]:= ?KnotTheoryVersion
KnotTheoryVersion[] returns the date of the current version of the package KnotTheory`. KnotTheoryVersion[k] returns the kth field in KnotTheoryVersion[].
In[5]:= ?KnotTheoryVersionString
KnotTheoryVersionString[] returns a string containing the date and time of the current version of the package KnotTheory`. It is generated from KnotTheoryVersion[].
In[6]:= ?KnotTheoryWelcomeMessage
KnotTheoryWelcomeMessage[] returns a string containing the welcome message printed when KnotTheory` is first loaded.

Thus on the day this manual page was last changed, we had:

In[7]:= {KnotTheoryVersion[], KnotTheoryVersionString[]}
Out[7]= {{2011, 3, 22, 21, 10, 4.67737}, March 22, 2011, 21:10:4.67737}
In[8]:= KnotTheoryWelcomeMessage[]
Out[8]= Loading KnotTheory` version of March 22, 2011, 21:10:4.67737. Read more at http://katlas.org/wiki/KnotTheory.
In[9]:= ?KnotTheoryDirectory
KnotTheoryDirectory[] returns the best guess KnotTheory` has for its location on the host computer. It can be reset by the user.
In[10]:= KnotTheoryDirectory[]
Out[10]= C:\Documents and Settings\pc\Documenti\Wolfram\KnotTheory

KnotTheoryDirectory may not work under some operating systems/environments. Please let Dror know if you encounter any difficulties.

Notes

Precomputed Data

KnotTheory` comes with a certain amount of precomputed data which is loaded "on demand" just when it is needed. When a precomputed data file is read by KnotTheory`, a notification message is displayed. To prevent these messages from appearing execute the command Off[KnotTheory::loading].

Further Data Files

To access the Hoste-Thistlethwaite enumeration of knots with 12 to 16 crossings (see Naming and Enumeration), also download either the file DTCodes4Knots12To16.tar.gz or the file DTCodes4Knots12To16.zip (about 9MB each), and unpack either one into the directory KnotTheory/.

Setting the Path

The directions above are written on the assumption that the package KnotTheory` (more precisely, the directory KnotTheory/ containing the files that make this package), is somewhere on your Mathematica search path. Usually this will be the case if KnotTheory/ is a subdirectory of your current working directory. If for some reason Mathematica cannot find KnotTheory`, you may tell it where to look in either of the following three ways. Assume KnotTheory/ is a subdirectory of FullPathToKnotTheory:

  1. If you are using KnotTheory` rarely and you don't want to change system defaults, evaluate AppendTo[$Path,"FullPathToKnotTheory"] within Mathematica before attempting to load KnotTheory`.
  2. If you plan to use KnotTheory` often, you may want to move the directory KnotTheory/ into one of the directories on your path. Evaluate $Path within Mathematica to see what those are.
  3. Alternatively, you may permanently add FullPathToKnotTheory to your $Path. To do that, find your Mathematica base directory by evaluating $UserBaseDirectory (on Dror's laptop, this comes out to be C:\Users\Dror\AppData\Roaming\Mathematica), and then add the line AppendTo[$Path,"FullPathToKnotTheory/"] to the file $BaseDirectory/Kernel/init.m and restart Mathematica.

Name 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 Knotscape):

(For In[1] see Setup)

In[2]:= ?Knot
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.
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.
6 1.gif
6_1 		9 46.gif
9_46

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]= 2 5 - - - 2 t t
In[5]:= Alexander[Knot[9, 46]][t]
Out[5]= 2 5 - - - 2 t t
L6a4.gif
L6a4

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

In[6]:= Length[Skeleton[Link[6, Alternating, 4]]]
Out[6]= 3
In[7]:= ?AllKnots
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.
In[8]:= ?AllLinks
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.

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

In[9]:= Length /@ {AllKnots[{0,16}], AllLinks[{2,12}]}
Out[9]= {1701936, 5700}
In[10]:= Show[DrawPD[Knot[13, NonAlternating, 5016], {Gap -> 0.025}]]
Naming and Enumeration Out 10.gif
Out[10]= -Graphics-

(Shumakovitch had noticed that this nice knot has interesting Khovanov homology; see [Shumakovitch]).

T(5,3).jpg
T(5,3)

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

In[11]:= ?TorusKnot
TorusKnot[m, n] represents the (m,n) torus knot.
In[12]:= ?TorusKnots
TorusKnots[n_] returns a list of all torus knots with up to n crossings.

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 ):

In[13]:= Crossings /@ {TorusKnot[5, 3], TorusKnot[3, 5]}
Out[13]= {10, 12}
In[14]:= Vassiliev[3] /@ {TorusKnot[5, 3], TorusKnot[3, 5]}
Out[14]= {20, 20}

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:

In[15]:= Knot /@ {"K11a14", "11a_14", "L8a1", "T(3,5)"}
Out[15]= {Knot[11, Alternating, 14], If[11 a <= 10 && 14 <= NumberOfKnots[11 a, Alternating] + NumberOfKnots[11 a, NonAlternating], Knot @@ KnotTheory`Naming`s$3008], Link[8, Alternating, 1], TorusKnot[3, 5]}

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

In[16]:= NameString /@ {Knot[11, Alternating, 14], TorusKnot[3,5]}
Out[16]= {K11a14, T(3,5)}

References

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

[Shumakovitch] ^  A. Shumakovitch, Torsion of the Khovanov Homology, arXiv:math.GT/0405474.

Presentations

KnotTheory` uses several presentations for knots/links.

Planar Diagrams

The PD notation

In the "Planar Diagrams" (PD) presentation we present every knot or link diagram by labeling its edges (with natural numbers, 1,...,n, and with increasing labels as we go around each component) and by a list crossings presented as symbols where , , and are the labels of the edges around that crossing, starting from the incoming lower edge and proceeding counterclockwise. Thus for example, the PD presentation of the knot above is:

(This of course is the Miller Institute knot, the mirror image of the knot 6_2)

(For In[1] see Setup)

In[2]:= ?PD
PD[v1, v2, ...] represents a planar diagram whose vertices are v1, v2, .... PD also acts as a "type caster", so for example, PD[K] where K is a named knot (or link) returns the PD presentation of that knot.
In[3]:= PD::about
The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
In[4]:= ?X
X[i,j,k,l] represents a crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counterclockwise through j, k and l. The (sometimes ambiguous) orientation of the upper strand is determined by the ordering of {j,l}.

Thus, for example, let us compute the determinant of the above knot:

In[5]:= K = PD[ X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,12], X[9,4,10,5], X[11,7,12,6] ];
In[6]:= Alexander[K][-1]
Out[6]= -11

Some further details

In[7]:= ?Xp
Xp[i,j,k,l] represents a positive (right handed) crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counter clockwise through j, k and l. The upper strand is therefore oriented from l to j regardless of the ordering of {j,l}. Presently Xp is only lightly supported.
In[8]:= ?Xm
Xm[i,j,k,l] represents a negative (left handed) crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counter clockwise through j, k and l. The upper strand is therefore oriented from j to l regardless of the ordering of {j,l}. Presently Xm is only lightly supported.
In[9]:= ?P
P[i,j] represents a bivalent vertex whose adjacent edges are i and j (i.e., a "Point" between the segment i and the segment j). Presently P is only lightly supported.

For example, we could add an extra "point" on the Miller Institute knot, splitting edge 12 into two pieces, labeled 12 and 13:

In[10]:= K1 = PD[ X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,13], X[9,4,10,5], X[11,7,12,6], P[12,13] ];

At the moment, many of our routines do not know to ignore such "extra points". But some do:

In[11]:= Jones[K][q] == Jones[K1][q]
Out[11]= True
In[12]:= ?Loop
Loop[i] represents a crossingsless loop labeled i.

Hence we can verify that the A2 invariant of the unknot is :

In[13]:= A2Invariant[Loop[1]][q]
Out[13]= -2 2 1 + q + q

Gauss Codes

The Gauss Code of an -crossing knot or link is obtained as follows:

  • Number the crossings of from 1 to in an arbitrary manner.
  • Order the components of is some arbitrary manner.
  • Start "walking" along the first component of , taking note of the numbers of the crossings you've gone through. If in a given crossing you cross on the "over" strand, write down the number of that crossing. If you cross on the "under" strand, write down the negative of the number of that crossing.
  • Do the same for all other components of (if any).

The resulting list of signed integers (in the case of a knot) or list of lists of signed integers (in the case of a link) is called the Gauss Code of . KnotTheory` has some rudimentary support for Gauss codes:

(For In[1] see Setup)

In[2]:= ?GaussCode
GaussCode[i1, i2, ...] represents a knot via its Gauss Code following the conventions used by the knotilus website, http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html. Likewise GaussCode[l1, l2, ...] represents a link, where each of l1, l2,... is a list describing the code read along one component of the link. GaussCode also acts as a "type caster", so for example, GaussCode[K] where K is is a named knot (or link) returns the Gauss code of that knot.

Thus for example, the Gauss codes for the trefoil knot and the Borromean link are:

In[3]:= GaussCode /@ {Knot[3, 1], Link[6, Alternating, 4]}
Out[3]= {GaussCode[-1, 3, -2, 1, -3, 2], GaussCode[{1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}]}
3 1.gif
3_1 		L6a4.gif
L6a4

Ralph Furmaniak, working under the guidance of Stuart Rankin and Ortho Flint at the University of Western Ontario, wrote a web-based server called "Knotilus" that takes Gauss codes and outputs pictures of the desired knots and links in several standard image formats.

In[4]:= ?KnotilusURL
KnotilusURL[K_] returns the URL of the knot/link K on the knotilus website, http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html.

Thus,

In[5]:= KnotilusURL /@ {Knot[3, 1], Link[6, Alternating, 4]}
Out[5]= {http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,3,-2,1,-3,2/goTop.h\ tml, http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,5,-3:4,-1,\ 2,-5:6,-4,3,-2/goTop.html}
Retrieved from "https://katlas.org/index.php?title=Printable_Manual&oldid=35930"