A Sample KnotTheory` Session: Difference between revisions
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{{InOut| |
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n = 21 | |
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in = <nowiki>{Vassiliev[2][K], Vassiliev[3][K]}</nowiki> | |
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out= <nowiki>{-1, 0}</nowiki>}} |
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{{InOut| |
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n = 22 | |
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in = <nowiki>Kh[TK][q, t]</nowiki> | |
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out= <nowiki> 23 25 27 2 31 3 29 4 31 4 33 5 35 5 |
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q + q + q t + q t + q t + q t + q t + q t + |
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31 6 33 6 35 7 37 7 33 8 35 8 37 9 |
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q t + q t + q t + q t + q t + 2 q t + q t + |
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39 9 37 10 41 11 39 12 41 12 45 12 |
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2 q t + 2 q t + 3 q t + q t + 2 q t + q t + |
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43 13 45 13 43 14 47 14 47 15 47 16 |
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2 q t + 2 q t + q t + q t + 2 q t + q t + |
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51 16 51 17 |
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q t + q t</nowiki>}} |
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n = 23 | |
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in = <nowiki>ColouredJones[K, &][q] /@ {2, 3, 4, 5, 6, 7}</nowiki> | |
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out= <nowiki>$Failed</nowiki>}} |
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Revision as of 15:46, 18 September 2005
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 |
 K11a231 |
 L8n6 |
 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]:=
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Show[BraidPlot[br]]
|
| |
Out[7]=
|
-Graphics-
|
In[8]:=
|
{First[br], Crossings[br], BraidIndex[K]}
|
Out[8]=
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{3, 8, 3}
|
In[9]:=
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Show[DrawMorseLink[K]]
|
| |
Out[9]=
|
-Graphics-
|
In[10]:=
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Show[DrawMorseLink[L]]
|
| |
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 |
 K11a108 |
 K11a139 |
K11a231 |
The Determinant and the Signature
In[15]:=
|
{KnotDet[K], KnotSignature[K]}
|
Out[15]=
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{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
|
Finite Type (Vassiliev) Invariants
In[21]:=
|
{Vassiliev[2][K], Vassiliev[3][K]}
|
Out[21]=
|
{-1, 0}
|
Khovanov Homology
In[22]:=
|
Kh[TK][q, t]
|
Out[22]=
|
23 25 27 2 31 3 29 4 31 4 33 5 35 5
q + q + q t + q t + q t + q t + q t + q t +
31 6 33 6 35 7 37 7 33 8 35 8 37 9
q t + q t + q t + q t + q t + 2 q t + q t +
39 9 37 10 41 11 39 12 41 12 45 12
2 q t + 2 q t + 3 q t + q t + 2 q t + q t +
43 13 45 13 43 14 47 14 47 15 47 16
2 q t + 2 q t + q t + q t + 2 q t + q t +
51 16 51 17
q t + q t
|
The Coloured Jones Polynomials
In[23]:=
|
ColouredJones[K, &][q] /@ {2, 3, 4, 5, 6, 7}
|
Out[23]=
|
$Failed
|
