A Sample KnotTheory` Session: Difference between revisions

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{{InOut|
{{InOut|
n = 18 |
n = 18 |
in = <nowiki>A2Invariant[K11][q]</nowiki> |
in = <nowiki>A2Invariant[L][q]</nowiki> |
out= <nowiki> -20 -18 3 -14 -12 6 -6 3 4 2
out= <nowiki> -30 3 4 5 4 3 2 -16 -14 -10 -8
5 + q - q + --- - q - q - -- + q - -- + -- + 2 q +
q + --- + --- + --- + --- + --- + --- + q + q + q + q +
16 8 4 2
28 26 24 22 20 18
q q q q
q q q q q q
⚫
-6
⚫
4 6 8 10 12
4 q - 2 q - q - q - q</nowiki>}}
q</nowiki>}}
<!--END-->
<!--END-->


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{{InOut|
{{InOut|
n = 19 |
n = 19 |
in = <nowiki>HOMFLYPT[K11][a, z]</nowiki> |
in = <nowiki>HOMFLYPT[L][a, z]</nowiki> |
out= <nowiki> 2
out= <nowiki> 6 8 10
4 2 4 2 4 z 2 2 4 2 4
4 8 a 2 a a 4 2 4 4
12 - -- - 10 a + 3 a + 18 z - ---- - 19 a z + 6 a z + 10 z -
2 a - 2 a + -- - ---- + --- + 4 a z + a z
2 2
2 2 2
a a
z z z</nowiki>}}
⚫
4
z 2 4 4 4 6 2 6 4 6 2 8
-- - 15 a z + 4 a z + 2 z - 6 a z + a z - a z
2
a</nowiki>}}
<!--END-->
<!--END-->


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{{InOut|
{{InOut|
n = 20 |
n = 20 |
in = <nowiki>Kauffman[K11][a, z]</nowiki> |
in = <nowiki>Kauffman[L][a, z]</nowiki> |
out= <nowiki> 4 2 4 4 z 11 z 3 5
out= <nowiki> 6 8 10 7 9
12 + -- + 10 a + 3 a - --- - ---- - 17 a z - 16 a z - 6 a z -
4 6 8 10 a 2 a a 2 a 2 a 7
2 3 a
2 a - 2 a - 9 a - 6 a + -- + ---- + --- - ---- - ---- + 8 a z +
a a
2 2 2 z z
⚫
z z z
2 3 3
2 8 z 2 2 4 2 6 2 8 2 8 z 20 z
24 z - ---- - 22 a z - 2 a z + 3 a z - a z + ---- + ----- +
2 3 a
a a
4
3 3 3 5 3 7 3 4 11 z 2 4
29 a z + 35 a z + 16 a z - 2 a z + 28 z + ----- + 30 a z +
2
a
5 5
4 4 6 4 8 4 5 z 9 z 5 3 5
6 a z - 6 a z + a z - ---- - ---- - 14 a z - 29 a z -
3 a
a
6
5 5 7 5 6 8 z 2 6 4 6 6 6
16 a z + 3 a z - 21 z - ---- - 31 a z - 12 a z + 6 a z +
2
a
7 7 8
9 4 2 8 2 10 2 7 3 9 3 4 4
z 3 z 7 3 7 5 7 8 2 z 2 8
8 a z - 4 a z + 14 a z + 10 a z - 6 a z - 6 a z + a z -
-- - ---- - 9 a z + 4 a z + 9 a z + 4 z + ---- + 10 a z +
3 a 2
a a
9
8 4 10 4 7 5 9 5 8 6 10 6
4 8 2 z 9 3 9 10 2 10
7 a z - 6 a z + a z + a z + a z + a z</nowiki>}}
8 a z + ---- + 6 a z + 4 a z + z + a z
a</nowiki>}}
<!--END-->
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Revision as of 15:47, 18 September 2005


KnotTheory`/Navigation 

The Mathematica Package KnotTheory`

Setup

The first step is to load KnotTheory` as in the Setup section:

In[1]:= << KnotTheory`

Loading KnotTheory` (version of September 14, 2005, 13:37:36)...

8 17.gif
8_17 		K11a231.gif
K11a231
L8n6.gif
L8n6 		T(7,5).jpg
T(7,5)

Let us now introduce the four star knots that will accompany us throughout this session:

In[2]:= K = Knot[8, 17]; K11 = Knot[11, Alternating, 231]; L = Link[8, NonAlternating, 6]; TK = TorusKnot[7,5];

Presentations

In[3]:= PD[K]
Out[3]= PD[X[6, 2, 7, 1], X[14, 8, 15, 7], X[8, 3, 9, 4], X[2, 13, 3, 14], X[12, 5, 13, 6], X[4, 9, 5, 10], X[16, 12, 1, 11], X[10, 16, 11, 15]]
In[4]:= {GaussCode[K], GaussCode[L]}
Out[4]= {GaussCode[1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 7, -5, 4, -2, 8, -7], GaussCode[{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}]}
In[5]:= DTCode[K]
Out[5]= DTCode[6, 8, 12, 14, 4, 16, 2, 10]
In[6]:= br = BR[K]
Out[6]= BR[3, {-1, -1, 2, -1, 2, -1, 2, 2}]
In[7]:= Show[BraidPlot[br]]
A Sample KnotTheory Session Out 7.gif
Out[7]= -Graphics-
In[8]:= {First[br], Crossings[br], BraidIndex[K]}
Out[8]= {3, 8, 3}
In[9]:= Show[DrawMorseLink[K]]
A Sample KnotTheory Session Out 9.gif
Out[9]= -Graphics-
In[10]:= Show[DrawMorseLink[L]]
A Sample KnotTheory Session Out 10.gif
Out[10]= -Graphics-

Three Dimensional Invariants

In[11]:= (#[K]&) /@ { SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex }
Out[11]= {NegativeAmphicheiral, 1, 3, 3, 4, 1}

Polynomial Invariants

The Alexander-Conway Polynomial

In[12]:= alex = Alexander[K11][t]
Out[12]= -4 5 12 20 2 3 4 -23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t 3 2 t t t
In[13]:= Conway[K11][t]
Out[13]= 2 4 6 8 1 + t - 2 t - 3 t - t
"Similar" Knots (within the Atlas)
In[14]:= Select[AllKnots[], (alex === Alexander[#][t])&]
Out[14]= {Knot[11, Alternating, 57], Knot[11, Alternating, 108], Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
K11a57.gif
K11a57 		K11a108.gif
K11a108 		K11a139.gif
K11a139 		K11a231.gif
K11a231
The Determinant and the Signature
In[15]:= {KnotDet[K], KnotSignature[K]}
Out[15]= {37, 0}

The Jones Polynomial

In[16]:= J=Jones[K11][q]
Out[16]= -7 3 7 12 14 16 16 2 3 4 -12 + q - -- + -- - -- + -- - -- + -- + 10 q - 5 q + 2 q - q 6 5 4 3 2 q q q q q q
"Similar" Knots (within the Atlas)
In[17]:= Select[AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q])&]
Out[17]= {Knot[11, Alternating, 57], Knot[11, Alternating, 231]}

The A2 Invariant

In[18]:= A2Invariant[L][q]
Out[18]= -30 3 4 5 4 3 2 -16 -14 -10 -8 q + --- + --- + --- + --- + --- + --- + q + q + q + q + 28 26 24 22 20 18 q q q q q q -6 q

The HOMFLY-PT Polynomial

In[19]:= HOMFLYPT[L][a, z]
Out[19]= 6 8 10 4 8 a 2 a a 4 2 4 4 2 a - 2 a + -- - ---- + --- + 4 a z + a z 2 2 2 z z z

The Kauffman Polynomial

In[20]:= Kauffman[L][a, z]
Out[20]= 6 8 10 7 9 4 6 8 10 a 2 a a 2 a 2 a 7 2 a - 2 a - 9 a - 6 a + -- + ---- + --- - ---- - ---- + 8 a z + 2 2 2 z z z z z 9 4 2 8 2 10 2 7 3 9 3 4 4 8 a z - 4 a z + 14 a z + 10 a z - 6 a z - 6 a z + a z - 8 4 10 4 7 5 9 5 8 6 10 6 7 a z - 6 a z + a z + a z + a z + a z
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