 819 |
 821 |
| © | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
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 KnotPlot |
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The Non Alternating Knot 820Also known as the "Oysterman's stopper" or as "Ashley's stopper knot". See 1. Visit 820's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 820's page at Knotilus! |
 KnotPlot |
| PD Presentation: | X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837 |
| Gauss Code: | {1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4} |
| DT (Dowker-Thistlethwaite) Code: | 4 8 -12 2 -14 -6 -16 -10 |
|
Minimum Braid Representative:
Length is 8, width is 3 Braid index is 3 |
A Morse Link Presentation:
|
| 3D Invariants: |
|
| Alexander Polynomial: | t-2 - 2t-1 + 3 - 2t + t2 |
| Conway Polynomial: | 1 + 2z2 + z4 |
| Other knots with the same Alexander/Conway Polynomial: | {10140, K11n73, K11n74, ...} |
| Determinant and Signature: | {9, 0} |
| Jones Polynomial: | - q-5 + q-4 - q-3 + 2q-2 - q-1 + 2 - q |
| Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
| A2 (sl(3)) Invariant: | - q-16 - q-14 - q-12 + 2q-8 + 2q-6 + 2q-4 + q-2 - q4 |
| HOMFLY-PT Polynomial: | - 1 - z2 + 4a2 + 4a2z2 + a2z4 - 2a4 - a4z2 |
| Kauffman Polynomial: | a-1z - 1 + 2z2 + 3az - 3az3 + az5 - 4a2 + 6a2z2 - 4a2z4 + a2z6 + 5a3z - 7a3z3 + 2a3z5 - 2a4 + 4a4z2 - 4a4z4 + a4z6 + 3a5z - 4a5z3 + a5z5 |
| V2 and V3, the type 2 and 3 Vassiliev invariants: | {2, -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=0 is the signature of 820. 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-15 - q-14 - q-13 + 2q-12 - q-11 - 2q-10 + 2q-9 - 2q-7 + 2q-6 + q-5 - 2q-4 + q-3 + 2q-2 - 2q-1 + 1 + q - q2 |
| 3 | - q-30 + q-29 + q-28 - 2q-26 + 2q-24 + q-23 - 2q-22 - q-21 + q-20 + q-19 - q-18 - q-17 + q-14 - 2q-11 + 2q-10 + q-9 - q-8 - 3q-7 + 4q-6 + q-5 - 2q-4 - 3q-3 + 4q-2 + 2q-1 - 1 - 3q + 2q2 + 2q3 - q4 - q5 - q6 + q7 |
| 4 | q-50 - q-49 - q-48 + 3q-45 - q-44 - q-43 - q-42 - 2q-41 + 4q-40 - 3q-36 + 3q-35 - q-34 + 2q-32 - 2q-31 + 2q-30 - 3q-29 - 2q-28 + 3q-27 + 4q-25 - 4q-24 - 5q-23 + 3q-22 + 2q-21 + 6q-20 - 4q-19 - 8q-18 + 2q-17 + 4q-16 + 7q-15 - 4q-14 - 10q-13 + 2q-12 + 5q-11 + 7q-10 - 4q-9 - 11q-8 + 2q-7 + 7q-6 + 7q-5 - 4q-4 - 11q-3 + 2q-2 + 6q-1 + 8 - 2q - 10q2 + 3q4 + 5q5 - 5q7 - q8 + 2q10 + q11 - q12 |
| 5 | - q-75 + q-74 + q-73 - q-70 - 2q-69 + 2q-67 + q-66 + q-65 - 2q-63 - 2q-62 + q-60 + q-59 + q-58 - q-56 - q-53 - q-52 + q-51 + q-50 + 3q-49 + q-48 - 2q-47 - 4q-46 - 2q-45 + 5q-43 + 5q-42 - 5q-40 - 6q-39 - 3q-38 + 4q-37 + 8q-36 + 5q-35 - 4q-34 - 8q-33 - 6q-32 + q-31 + 10q-30 + 8q-29 - q-28 - 9q-27 - 9q-26 - 2q-25 + 10q-24 + 11q-23 + q-22 - 9q-21 - 11q-20 - 5q-19 + 11q-18 + 13q-17 + 2q-16 - 10q-15 - 13q-14 - 5q-13 + 13q-12 + 14q-11 + 2q-10 - 11q-9 - 13q-8 - 5q-7 + 13q-6 + 15q-5 + 3q-4 - 10q-3 - 13q-2 - 6q-1 + 10 + 14q + 5q2 - 7q3 - 11q4 - 6q5 + 4q6 + 8q7 + 5q8 - 2q9 - 6q10 - 2q11 + 2q13 + 2q14 - q16 |
| 6 | q-105 - q-104 - q-103 + q-100 + 3q-98 - q-97 - 2q-96 - q-95 - q-94 - q-92 + 5q-91 - q-87 - q-86 - 4q-85 + 4q-84 - q-83 + q-81 + q-80 + 2q-79 - 3q-78 + 3q-77 - 3q-76 - 3q-75 - 2q-74 + 5q-72 + q-71 + 6q-70 - q-69 - 3q-68 - 6q-67 - 6q-66 + 3q-65 + q-64 + 9q-63 + 5q-62 + 2q-61 - 6q-60 - 10q-59 - 2q-58 - 5q-57 + 7q-56 + 9q-55 + 8q-54 - q-53 - 9q-52 - 4q-51 - 12q-50 + q-49 + 9q-48 + 12q-47 + 5q-46 - 4q-45 - 2q-44 - 17q-43 - 6q-42 + 5q-41 + 13q-40 + 10q-39 + q-38 + 2q-37 - 20q-36 - 13q-35 + q-34 + 14q-33 + 14q-32 + 5q-31 + 5q-30 - 23q-29 - 19q-28 - q-27 + 15q-26 + 17q-25 + 8q-24 + 5q-23 - 25q-22 - 22q-21 + 17q-19 + 18q-18 + 8q-17 + 4q-16 - 26q-15 - 23q-14 + q-13 + 19q-12 + 19q-11 + 8q-10 + 4q-9 - 27q-8 - 24q-7 + 18q-5 + 20q-4 + 10q-3 + 7q-2 - 25q-1 - 24 - 4q + 13q2 + 17q3 + 11q4 + 10q5 - 17q6 - 19q7 - 6q8 + 5q9 + 8q10 + 7q11 + 9q12 - 6q13 - 7q14 - 4q15 + q18 + 5q19 - q20 - q24 - q25 + q26 |
| 7 | - q-140 + q-139 + q-138 - q-135 - q-133 - 2q-132 + q-131 + 2q-130 + q-129 + 2q-128 - q-127 - 4q-124 - q-123 + 3q-120 + q-118 + 3q-117 - 2q-116 - q-114 - 3q-113 + 2q-112 - 2q-111 - 2q-110 + 2q-109 + 3q-107 + 3q-106 - q-105 + 4q-104 - 2q-103 - 6q-102 - 3q-101 - 5q-100 + q-99 + 4q-98 + 2q-97 + 9q-96 + 5q-95 - 3q-94 - 3q-93 - 9q-92 - 6q-91 - 3q-90 - 2q-89 + 9q-88 + 10q-87 + 5q-86 + 5q-85 - 3q-84 - 8q-83 - 9q-82 - 10q-81 + 6q-79 + 6q-78 + 13q-77 + 7q-76 - 7q-74 - 13q-73 - 10q-72 - 5q-71 + 11q-69 + 14q-68 + 11q-67 + 4q-66 - 8q-65 - 14q-64 - 15q-63 - 10q-62 + 3q-61 + 13q-60 + 19q-59 + 16q-58 + 3q-57 - 13q-56 - 21q-55 - 20q-54 - 8q-53 + 10q-52 + 22q-51 + 26q-50 + 14q-49 - 9q-48 - 25q-47 - 28q-46 - 17q-45 + 6q-44 + 24q-43 + 33q-42 + 23q-41 - 7q-40 - 28q-39 - 34q-38 - 24q-37 + 5q-36 + 27q-35 + 38q-34 + 28q-33 - 7q-32 - 31q-31 - 37q-30 - 28q-29 + 6q-28 + 30q-27 + 40q-26 + 30q-25 - 9q-24 - 33q-23 - 38q-22 - 28q-21 + 9q-20 + 31q-19 + 40q-18 + 29q-17 - 11q-16 - 34q-15 - 38q-14 - 28q-13 + 10q-12 + 32q-11 + 42q-10 + 30q-9 - 10q-8 - 35q-7 - 39q-6 - 31q-5 + 6q-4 + 31q-3 + 44q-2 + 34q-1 - 3 - 30q - 38q2 - 33q3 - 3q4 + 21q5 + 35q6 + 34q7 + 6q8 - 16q9 - 26q10 - 28q11 - 9q12 + 6q13 + 16q14 + 21q15 + 11q16 - 10q18 - 14q19 - 7q20 - q21 + 7q23 + 6q24 + 4q25 - 3q27 - 2q28 - q29 - 2q30 + q32 + q33 + q34 - q35 |
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[8, 20]] |
Out[2]= | PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[13, 16, 14, 1], > X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[2, 8, 3, 7]] |
In[3]:= | GaussCode[Knot[8, 20]] |
Out[3]= | GaussCode[1, -8, 2, -1, -3, 7, 8, -2, -5, 6, -7, 3, -4, 5, -6, 4] |
In[4]:= | DTCode[Knot[8, 20]] |
Out[4]= | DTCode[4, 8, -12, 2, -14, -6, -16, -10] |
In[5]:= | br = BR[Knot[8, 20]] |
Out[5]= | BR[3, {1, 1, 1, -2, -1, -1, -1, -2}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {3, 8} |
In[7]:= | BraidIndex[Knot[8, 20]] |
Out[7]= | 3 |
In[8]:= | Show[DrawMorseLink[Knot[8, 20]]] |
| |
Out[8]= | -Graphics- |
In[9]:= | #[Knot[8, 20]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 1, 2, 3, 4, 1} |
In[10]:= | alex = Alexander[Knot[8, 20]][t] |
Out[10]= | -2 2 2
3 + t - - - 2 t + t
t |
In[11]:= | Conway[Knot[8, 20]][z] |
Out[11]= | 2 4 1 + 2 z + z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[8, 20], Knot[10, 140], Knot[11, NonAlternating, 73],
> Knot[11, NonAlternating, 74]} |
In[13]:= | {KnotDet[Knot[8, 20]], KnotSignature[Knot[8, 20]]} |
Out[13]= | {9, 0} |
In[14]:= | Jones[Knot[8, 20]][q] |
Out[14]= | -5 -4 -3 2 1
2 - q + q - q + -- - - - q
2 q
q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[8, 20]} |
In[16]:= | A2Invariant[Knot[8, 20]][q] |
Out[16]= | -16 -14 -12 2 2 2 -2 4
-q - q - q + -- + -- + -- + q - q
8 6 4
q q q |
In[17]:= | HOMFLYPT[Knot[8, 20]][a, z] |
Out[17]= | 2 4 2 2 2 4 2 2 4 -1 + 4 a - 2 a - z + 4 a z - a z + a z |
In[18]:= | Kauffman[Knot[8, 20]][a, z] |
Out[18]= | 2 4 z 3 5 2 2 2 4 2
-1 - 4 a - 2 a + - + 3 a z + 5 a z + 3 a z + 2 z + 6 a z + 4 a z -
a
3 3 3 5 3 2 4 4 4 5 3 5 5 5
> 3 a z - 7 a z - 4 a z - 4 a z - 4 a z + a z + 2 a z + a z +
2 6 4 6
> a z + a z |
In[19]:= | {Vassiliev[2][Knot[8, 20]], Vassiliev[3][Knot[8, 20]]} |
Out[19]= | {2, -2} |
In[20]:= | Kh[Knot[8, 20]][q, t] |
Out[20]= | 2 1 1 1 1 1 1 3
- + q + ------ + ----- + ----- + ----- + ----- + --- + q t
q 11 5 7 4 7 3 5 2 3 2 q t
q t q t q t q t q t |
In[21]:= | ColouredJones[Knot[8, 20], 2][q] |
Out[21]= | -15 -14 -13 2 -11 2 2 2 2 -5 2 -3
1 + q - q - q + --- - q - --- + -- - -- + -- + q - -- + q +
12 10 9 7 6 4
q q q q q q
2 2 2
> -- - - + q - q
2 q
q |
| Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 820 |
|
