A Sample KnotTheory` Session: Difference between revisions

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Line 216: Line 216:
n = 19 |
n = 19 |
in = <nowiki>HOMFLYPT[K11][a, z]</nowiki> |
in = <nowiki>HOMFLYPT[K11][a, z]</nowiki> |
⚫
out= <nowiki> 2
out= <nowiki>Function[{KnotTheory`HOMFLYPT`a, KnotTheory`HOMFLYPT`z},
⚫
4 2 4 2 4 z 2 2 4 2 4
⚫
12 - -- - 10 a + 3 a + 18 z - ---- - 19 a z + 6 a z + 10 z -
⚫
2 2
⚫
a a
⚫
4
⚫
4 2
⚫
z 2 4 4 4 6 2 6 4 6 2 8
12 - ---------------------- - 10 KnotTheory`HOMFLYPT`a +
2
-- - 15 a z + 4 a z + 2 z - 6 a z + a z - a z
2
KnotTheory`HOMFLYPT`a
a</nowiki>}}
4 2
3 KnotTheory`HOMFLYPT`a + 18 KnotTheory`HOMFLYPT`z -
⚫
2
4 KnotTheory`HOMFLYPT`z
------------------------ -
2
KnotTheory`HOMFLYPT`a
2 2
19 KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z +
4 2
6 KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z +
⚫
4
4 KnotTheory`HOMFLYPT`z
10 KnotTheory`HOMFLYPT`z - ---------------------- -
⚫
2
KnotTheory`HOMFLYPT`a
⚫
2 4
15 KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z +
4 4
4 KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z +
⚫
6
2 KnotTheory`HOMFLYPT`z -
2 6
6 KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z +
⚫
4 6
KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z -
⚫
2 8
KnotTheory`HOMFLYPT`a KnotTheory`HOMFLYPT`z ][q]</nowiki>}}
<!--END-->
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Line 269: Line 235:
{{InOut|
{{InOut|
n = 20 |
n = 20 |
in = <nowiki>Kauffman[K11][q]</nowiki> |
in = <nowiki>Kauffman[K11][a, z]</nowiki> |
out= <nowiki> 4 2 4 4 #2 11 #2 3 5
out= <nowiki> 4 2 4 4 z 11 z 3 5
12 + -- + 10 q + 3 q - ---- - ----- - 17 q #2 - 16 q #2 - 6 q #2 -
12 + -- + 10 a + 3 a - --- - ---- - 17 a z - 16 a z - 6 a z -
2 3 q
2 3 a
q q
a a
⚫
2 3
2 8 #2 2 2 4 2 6 2 8 2 8 #2
⚫
24 #2 - ----- - 22 q #2 - 2 q #2 + 3 q #2 - q #2 + ----- +
2 3
q q
⚫
2 3 3
3
20 #2 3 3 3 5 3 7 3 4
2 8 z 2 2 4 2 6 2 8 2 8 z 20 z
------ + 29 q #2 + 35 q #2 + 16 q #2 - 2 q #2 + 28 #2 +
24 z - ---- - 22 a z - 2 a z + 3 a z - a z + ---- + ----- +
⚫
2 3 a
⚫
q
⚫
a a
4 5 5
4
11 #2 2 4 4 4 6 4 8 4 5 #2 9 #2
3 3 3 5 3 7 3 4 11 z 2 4
------ + 30 q #2 + 6 q #2 - 6 q #2 + q #2 - ----- - ----- -
29 a z + 35 a z + 16 a z - 2 a z + 28 z + ----- + 30 a z +
2 3 q
2
q q
a
6
5 5
5 3 5 5 5 7 5 6 8 #2
4 4 6 4 8 4 5 z 9 z 5 3 5
14 q #2 - 29 q #2 - 16 q #2 + 3 q #2 - 21 #2 - ----- -
6 a z - 6 a z + a z - ---- - ---- - 14 a z - 29 a z -
2
3 a
q
a
7 7
6
2 6 4 6 6 6 #2 3 #2 7 3 7
5 5 7 5 6 8 z 2 6 4 6 6 6
31 q #2 - 12 q #2 + 6 q #2 + --- - ----- - 9 q #2 + 4 q #2 +
16 a z + 3 a z - 21 z - ---- - 31 a z - 12 a z + 6 a z +
3 q
2
q
a
8 9
7 7 8
5 7 8 2 #2 2 8 4 8 2 #2 9
z 3 z 7 3 7 5 7 8 2 z 2 8
9 q #2 + 4 #2 + ----- + 10 q #2 + 8 q #2 + ----- + 6 q #2 +
-- - ---- - 9 a z + 4 a z + 9 a z + 4 z + ---- + 10 a z +
2 q
3 a 2
q
a a
3 9 10 2 10
9
4 q #2 + #2 + q #2</nowiki>}}
4 8 2 z 9 3 9 10 2 10
⚫
8 a z + ---- + 6 a z + 4 a z + z + a z
⚫
a</nowiki>}}
<!--END-->
<!--END-->

Revision as of 15:44, 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[K11][q]
Out[18]= -20 -18 3 -14 -12 6 -6 3 4 2 5 + q - q + --- - q - q - -- + q - -- + -- + 2 q + 16 8 4 2 q q q q 4 6 8 10 12 4 q - 2 q - q - q - q

The HOMFLY-PT Polynomial

In[19]:= HOMFLYPT[K11][a, z]
Out[19]= 2 4 2 4 2 4 z 2 2 4 2 4 12 - -- - 10 a + 3 a + 18 z - ---- - 19 a z + 6 a z + 10 z - 2 2 a a 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

The Kauffman Polynomial

In[20]:= Kauffman[K11][a, z]
Out[20]= 4 2 4 4 z 11 z 3 5 12 + -- + 10 a + 3 a - --- - ---- - 17 a z - 16 a z - 6 a z - 2 3 a a a 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 z 3 z 7 3 7 5 7 8 2 z 2 8 -- - ---- - 9 a z + 4 a z + 9 a z + 4 z + ---- + 10 a z + 3 a 2 a a 9 4 8 2 z 9 3 9 10 2 10 8 a z + ---- + 6 a z + 4 a z + z + a z a
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