 1073 |
 1075 |
| © | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
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The Alternating Knot 1074Visit 1074's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1074's page at Knotilus! |
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| PD Presentation: | X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,18,12,19 X9,20,10,1 X19,10,20,11 X17,6,18,7 X7,16,8,17 X15,8,16,9 |
| Gauss Code: | {-1, 4, -3, 1, -2, 8, -9, 10, -6, 7, -5, 3, -4, 2, -10, 9, -8, 5, -7, 6} |
| DT (Dowker-Thistlethwaite) Code: | 4 12 14 16 20 18 2 8 6 10 |
|
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: | {1067, K11n68, ...} |
| Determinant and Signature: | {63, -2} |
| Jones Polynomial: | q-9 - 2q-8 + 4q-7 - 8q-6 + 9q-5 - 10q-4 + 11q-3 - 8q-2 + 6q-1 - 3 + q |
| Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
| A2 (sl(3)) Invariant: | q-28 + 2q-22 - 3q-20 - 2q-18 - 2q-14 + 2q-12 + 2q-8 + 2q-6 - q-4 + 3q-2 - 1 - q2 + q4 |
| HOMFLY-PT Polynomial: | z2 + 2a2 + a2z2 - a2z4 - 2a4z2 - 2a4z4 - 2a6 - a6z2 - a6z4 + a8 + a8z2 |
| Kauffman Polynomial: | - z2 + z4 - 3az3 + 3az5 - 2a2 + 5a2z2 - 7a2z4 + 5a2z6 + 3a3z3 - 6a3z5 + 5a3z7 + 8a4z2 - 9a4z4 + a4z6 + 3a4z8 - 4a5z + 9a5z3 - 12a5z5 + 5a5z7 + a5z9 + 2a6 - a6z2 + 3a6z4 - 9a6z6 + 5a6z8 - 8a7z + 11a7z3 - 10a7z5 + 2a7z7 + a7z9 + a8 + a8z2 - 4a8z6 + 2a8z8 - 4a9z + 8a9z3 - 7a9z5 + 2a9z7 + 4a10z2 - 4a10z4 + a10z6 |
| V2 and V3, the type 2 and 3 Vassiliev invariants: | {0, 2} |
|
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 1074. 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 - 2q-25 + 6q-23 - 9q-22 - 4q-21 + 23q-20 - 19q-19 - 19q-18 + 53q-17 - 25q-16 - 47q-15 + 79q-14 - 19q-13 - 75q-12 + 90q-11 - 6q-10 - 86q-9 + 82q-8 + 6q-7 - 72q-6 + 54q-5 + 11q-4 - 43q-3 + 24q-2 + 7q-1 - 16 + 7q + 2q2 - 3q3 + 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, 74]] |
Out[2]= | PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[11, 18, 12, 19], X[9, 20, 10, 1], X[19, 10, 20, 11], X[17, 6, 18, 7], > X[7, 16, 8, 17], X[15, 8, 16, 9]] |
In[3]:= | GaussCode[Knot[10, 74]] |
Out[3]= | GaussCode[-1, 4, -3, 1, -2, 8, -9, 10, -6, 7, -5, 3, -4, 2, -10, 9, -8, 5, -7, > 6] |
In[4]:= | DTCode[Knot[10, 74]] |
Out[4]= | DTCode[4, 12, 14, 16, 20, 18, 2, 8, 6, 10] |
In[5]:= | br = BR[Knot[10, 74]] |
Out[5]= | BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, 2, 4, -3, -2, 4, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {5, 14} |
In[7]:= | BraidIndex[Knot[10, 74]] |
Out[7]= | 5 |
In[8]:= | Show[DrawMorseLink[Knot[10, 74]]] |
| |
Out[8]= | -Graphics- |
In[9]:= | #[Knot[10, 74]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 2, 2, 3, NotAvailable, 2} |
In[10]:= | alex = Alexander[Knot[10, 74]][t] |
Out[10]= | 4 16 2
-23 - -- + -- + 16 t - 4 t
2 t
t |
In[11]:= | Conway[Knot[10, 74]][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, 74]], KnotSignature[Knot[10, 74]]} |
Out[13]= | {63, -2} |
In[14]:= | Jones[Knot[10, 74]][q] |
Out[14]= | -9 2 4 8 9 10 11 8 6
-3 + 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, 74]} |
In[16]:= | A2Invariant[Knot[10, 74]][q] |
Out[16]= | -28 2 3 2 2 2 2 2 -4 3 2 4
-1 + q + --- - --- - --- - --- + --- + -- + -- - q + -- - q + q
22 20 18 14 12 8 6 2
q q q q q q q q |
In[17]:= | HOMFLYPT[Knot[10, 74]][a, z] |
Out[17]= | 2 6 8 2 2 2 4 2 6 2 8 2 2 4 4 4
2 a - 2 a + a + z + a z - 2 a z - a z + a z - a z - 2 a z -
6 4
> a z |
In[18]:= | Kauffman[Knot[10, 74]][a, z] |
Out[18]= | 2 6 8 5 7 9 2 2 2 4 2 6 2
-2 a + 2 a + a - 4 a z - 8 a z - 4 a z - z + 5 a z + 8 a z - a z +
8 2 10 2 3 3 3 5 3 7 3 9 3 4
> a z + 4 a z - 3 a z + 3 a z + 9 a z + 11 a z + 8 a z + z -
2 4 4 4 6 4 10 4 5 3 5 5 5
> 7 a z - 9 a z + 3 a z - 4 a z + 3 a z - 6 a z - 12 a z -
7 5 9 5 2 6 4 6 6 6 8 6 10 6
> 10 a z - 7 a z + 5 a z + a z - 9 a z - 4 a z + a z +
3 7 5 7 7 7 9 7 4 8 6 8 8 8
> 5 a z + 5 a z + 2 a z + 2 a z + 3 a z + 5 a z + 2 a z +
5 9 7 9
> a z + a z |
In[19]:= | {Vassiliev[2][Knot[10, 74]], Vassiliev[3][Knot[10, 74]]} |
Out[19]= | {0, 2} |
In[20]:= | Kh[Knot[10, 74]][q, t] |
Out[20]= | 3 4 1 1 1 3 1 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
4 5 6 4 5 6 3 5 t
> ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 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, 74], 2][q] |
Out[21]= | -26 2 6 9 4 23 19 19 53 25 47 79
-16 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- -
25 23 22 21 20 19 18 17 16 15 14
q q q q q q q q q q q
19 75 90 6 86 82 6 72 54 11 43 24 7
> --- - --- + --- - --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 7 q +
13 12 11 10 9 8 7 6 5 4 3 2 q
q q q q q q q q q q q q
2 3 4
> 2 q - 3 q + q |
| Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1074 |
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