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[L][q]
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Out[18]=
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-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
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In[19]:=
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HOMFLYPT[L][a, z]
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Out[19]=
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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
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In[20]:=
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Kauffman[L][a, z]
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Out[20]=
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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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