 1066 |
 1068 |
| © | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
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The Alternating Knot 1067Visit 1067's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1067's page at Knotilus! |
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| PD Presentation: | X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X9,18,10,19 X15,20,16,1 X19,16,20,17 X17,8,18,9 |
| Gauss Code: | {-1, 4, -3, 1, -5, 6, -2, 10, -7, 3, -4, 2, -6, 5, -8, 9, -10, 7, -9, 8} |
| DT (Dowker-Thistlethwaite) Code: | 4 10 14 12 18 2 6 20 8 16 |
|
Minimum Braid Representative:
Length is 14, width is 5 Braid index is 5 |
A Morse Link Presentation:
|
| 3D Invariants: |
|
| Alexander Polynomial: | - 4t-2 + 16t-1 - 23 + 16t - 4t2 |
| Conway Polynomial: | 1 - 4z4 |
| Other knots with the same Alexander/Conway Polynomial: | {1074, K11n68, ...} |
| Determinant and Signature: | {63, -2} |
| Jones Polynomial: | q-9 - 3q-8 + 5q-7 - 8q-6 + 10q-5 - 10q-4 + 10q-3 - 8q-2 + 5q-1 - 2 + q |
| Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
| A2 (sl(3)) Invariant: | q-28 - q-26 - q-24 + 2q-22 - 2q-20 + q-16 - q-14 + 2q-12 - q-10 + q-8 - 2q-4 + 3q-2 + q4 |
| HOMFLY-PT Polynomial: | 1 + z2 - a2z4 - 2a4z2 - 2a4z4 - a6z4 + a8z2 |
| Kauffman Polynomial: | 1 - 2z2 + z4 - 2az3 + 2az5 - 2a2z4 + 3a2z6 - 2a3z + 7a3z3 - 6a3z5 + 4a3z7 + 2a4z2 - a4z4 - 2a4z6 + 3a4z8 - 6a5z + 19a5z3 - 19a5z5 + 6a5z7 + a5z9 - 2a6z2 + 7a6z4 - 13a6z6 + 6a6z8 - 6a7z + 19a7z3 - 21a7z5 + 5a7z7 + a7z9 + 2a8z4 - 7a8z6 + 3a8z8 - 2a9z + 9a9z3 - 10a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6 |
| V2 and V3, the type 2 and 3 Vassiliev invariants: | {0, 0} |
|
Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-2 is the signature of 1067. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.) |
|
| n | Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2)) |
| 2 | q-26 - 3q-25 + 10q-23 - 12q-22 - 8q-21 + 32q-20 - 19q-19 - 31q-18 + 58q-17 - 15q-16 - 61q-15 + 75q-14 + q-13 - 83q-12 + 75q-11 + 17q-10 - 86q-9 + 58q-8 + 23q-7 - 64q-6 + 33q-5 + 18q-4 - 32q-3 + 13q-2 + 7q-1 - 10 + 4q + q2 - 2q3 + q4 |
Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... | |
In[2]:= | PD[Knot[10, 67]] |
Out[2]= | PD[X[1, 4, 2, 5], X[7, 12, 8, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[5, 14, 6, 15], X[13, 6, 14, 7], X[9, 18, 10, 19], X[15, 20, 16, 1], > X[19, 16, 20, 17], X[17, 8, 18, 9]] |
In[3]:= | GaussCode[Knot[10, 67]] |
Out[3]= | GaussCode[-1, 4, -3, 1, -5, 6, -2, 10, -7, 3, -4, 2, -6, 5, -8, 9, -10, 7, -9, > 8] |
In[4]:= | DTCode[Knot[10, 67]] |
Out[4]= | DTCode[4, 10, 14, 12, 18, 2, 6, 20, 8, 16] |
In[5]:= | br = BR[Knot[10, 67]] |
Out[5]= | BR[5, {-1, -1, -1, -2, 1, -2, -3, 2, 2, 4, -3, -2, 4, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {5, 14} |
In[7]:= | BraidIndex[Knot[10, 67]] |
Out[7]= | 5 |
In[8]:= | Show[DrawMorseLink[Knot[10, 67]]] |
| |
Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 67]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Chiral, 2, 2, 3, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 67]][t] |
Out[10]= | 4 16 2
-23 - -- + -- + 16 t - 4 t
2 t
t |
In[11]:= | Conway[Knot[10, 67]][z] |
Out[11]= | 4 1 - 4 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 67], Knot[10, 74], Knot[11, NonAlternating, 68]} |
In[13]:= | {KnotDet[Knot[10, 67]], KnotSignature[Knot[10, 67]]} |
Out[13]= | {63, -2} |
In[14]:= | Jones[Knot[10, 67]][q] |
Out[14]= | -9 3 5 8 10 10 10 8 5
-2 + q - -- + -- - -- + -- - -- + -- - -- + - + q
8 7 6 5 4 3 2 q
q q q q q q q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 67]} |
In[16]:= | A2Invariant[Knot[10, 67]][q] |
Out[16]= | -28 -26 -24 2 2 -16 -14 2 -10 -8 2 3 4
q - q - q + --- - --- + q - q + --- - q + q - -- + -- + q
22 20 12 4 2
q q q q q |
In[17]:= | HOMFLYPT[Knot[10, 67]][a, z] |
Out[17]= | 2 4 2 8 2 2 4 4 4 6 4 1 + z - 2 a z + a z - a z - 2 a z - a z |
In[18]:= | Kauffman[Knot[10, 67]][a, z] |
Out[18]= | 3 5 7 9 2 4 2 6 2 10 2
1 - 2 a z - 6 a z - 6 a z - 2 a z - 2 z + 2 a z - 2 a z + 2 a z -
3 3 3 5 3 7 3 9 3 4 2 4 4 4
> 2 a z + 7 a z + 19 a z + 19 a z + 9 a z + z - 2 a z - a z +
6 4 8 4 10 4 5 3 5 5 5 7 5
> 7 a z + 2 a z - 3 a z + 2 a z - 6 a z - 19 a z - 21 a z -
9 5 2 6 4 6 6 6 8 6 10 6 3 7
> 10 a z + 3 a z - 2 a z - 13 a z - 7 a z + a z + 4 a z +
5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9
> 6 a z + 5 a z + 3 a z + 3 a z + 6 a z + 3 a z + a z + a z |
In[19]:= | {Vassiliev[2][Knot[10, 67]], Vassiliev[3][Knot[10, 67]]} |
Out[19]= | {0, 0} |
In[20]:= | Kh[Knot[10, 67]][q, t] |
Out[20]= | 2 4 1 2 1 3 2 5 3
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ +
3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5
q q t q t q t q t q t q t q t
5 5 5 5 5 5 3 5 t
> ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t +
11 4 9 4 9 3 7 3 7 2 5 2 5 3 q
q t q t q t q t q t q t q t q t
3 2
> q t |
In[21]:= | ColouredJones[Knot[10, 67], 2][q] |
Out[21]= | -26 3 10 12 8 32 19 31 58 15 61 75
-10 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- +
25 23 22 21 20 19 18 17 16 15 14
q q q q q q q q q q q
-13 83 75 17 86 58 23 64 33 18 32 13 7
> q - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 4 q +
12 11 10 9 8 7 6 5 4 3 2 q
q q q q q q q q q q q
2 3 4
> q - 2 q + q |
| Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1067 |
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