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)...
Let us now introduce the four star knots that will accompany us throughout this session:
In[2]:=
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K = Knot[8, 17];
K11 = Knot[11, Alternating, 231];
L = Link[8, NonAlternating, 6];
TK = TorusKnot[7,5];
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In[3]:=
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PD[K]
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Out[3]=
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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]]
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In[4]:=
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{GaussCode[K], GaussCode[L]}
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Out[4]=
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{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}]}
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In[5]:=
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DTCode[K]
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Out[5]=
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DTCode[6, 8, 12, 14, 4, 16, 2, 10]
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In[6]:=
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br = BR[K]
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Out[6]=
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BR[3, {-1, -1, 2, -1, 2, -1, 2, 2}]
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In[7]:=
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Show[BraidPlot[br]]
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Out[7]=
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-Graphics-
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In[8]:=
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{First[br], Crossings[br], BraidIndex[K]}
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Out[8]=
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{3, 8, 3}
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In[9]:=
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Show[DrawMorseLink[K]]
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Out[9]=
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-Graphics-
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In[10]:=
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Show[DrawMorseLink[L]]
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Out[10]=
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-Graphics-
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In[11]:=
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(#[K]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}
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Out[11]=
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{NegativeAmphicheiral, 1, 3, 3, 4, 1}
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Polynomial Invariants
In[12]:=
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alex = Alexander[K11][t]
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Out[12]=
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-4 5 12 20 2 3 4
-23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t
3 2 t
t t
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In[13]:=
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Conway[K11][t]
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Out[13]=
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2 4 6 8
1 + t - 2 t - 3 t - t
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"Similar" Knots (within the Atlas)
In[14]:=
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Select[AllKnots[], (alex === Alexander[#][t])&]
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Out[14]=
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{Knot[11, Alternating, 57], Knot[11, Alternating, 108],
Knot[11, Alternating, 139], Knot[11, Alternating, 231]}
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In[15]:=
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{KnotDet[K], KnotSignature[K]}
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Out[15]=
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{37, 0}
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In[16]:=
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J=Jones[K11][q]
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Out[16]=
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-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
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"Similar" Knots (within the Atlas)
In[17]:=
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Select[AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q])&]
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Out[17]=
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{Knot[11, Alternating, 57], Knot[11, Alternating, 231]}
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In[18]:=
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A2Invariant[K11][q]
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Out[18]=
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-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
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In[19]:=
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HOMFLYPT[K11][a, z]
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Out[19]=
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Function[{KnotTheory`HOMFLYPT`a, KnotTheory`HOMFLYPT`z},
4 2
12 - ---------------------- - 10 KnotTheory`HOMFLYPT`a +
2
KnotTheory`HOMFLYPT`a
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]
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In[20]:=
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Kauffman[K11][q]
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Out[20]=
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4 2 4 4 #2 11 #2 3 5
12 + -- + 10 q + 3 q - ---- - ----- - 17 q #2 - 16 q #2 - 6 q #2 -
2 3 q
q q
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
3
20 #2 3 3 3 5 3 7 3 4
------ + 29 q #2 + 35 q #2 + 16 q #2 - 2 q #2 + 28 #2 +
q
4 5 5
11 #2 2 4 4 4 6 4 8 4 5 #2 9 #2
------ + 30 q #2 + 6 q #2 - 6 q #2 + q #2 - ----- - ----- -
2 3 q
q q
6
5 3 5 5 5 7 5 6 8 #2
14 q #2 - 29 q #2 - 16 q #2 + 3 q #2 - 21 #2 - ----- -
2
q
7 7
2 6 4 6 6 6 #2 3 #2 7 3 7
31 q #2 - 12 q #2 + 6 q #2 + --- - ----- - 9 q #2 + 4 q #2 +
3 q
q
8 9
5 7 8 2 #2 2 8 4 8 2 #2 9
9 q #2 + 4 #2 + ----- + 10 q #2 + 8 q #2 + ----- + 6 q #2 +
2 q
q
3 9 10 2 10
4 q #2 + #2 + q #2
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