 945 |
 947 |
| © | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: |
|
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The Non Alternating Knot 946Visit 946's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 946's page at Knotilus! |
 KnotPlot |
| PD Presentation: | X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8 |
| Gauss Code: | {1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7} |
| DT (Dowker-Thistlethwaite) Code: | 4 10 -14 -12 -16 2 -6 -18 -8 |
|
Minimum Braid Representative:
Length is 9, width is 4 Braid index is 4 |
A Morse Link Presentation:
|
| 3D Invariants: |
|
| Alexander Polynomial: | - 2t-1 + 5 - 2t |
| Conway Polynomial: | 1 - 2z2 |
| Other knots with the same Alexander/Conway Polynomial: | {61, K11n67, K11n97, K11n139, ...} |
| Determinant and Signature: | {9, 0} |
| Jones Polynomial: | q-6 - q-5 + q-4 - 2q-3 + q-2 - q-1 + 2 |
| Other knots (up to mirrors) with the same Jones Polynomial: | {...} |
| A2 (sl(3)) Invariant: | q-20 + q-18 - q-12 - q-10 - q-8 - q-6 + q-2 + 2 + 2q2 |
| HOMFLY-PT Polynomial: | 2 - a2 - a2z2 - a4 - a4z2 + a6 |
| Kauffman Polynomial: | 2 - 2az + az3 + a2 + 3a2z2 - 4a2z4 + a2z6 - 6a3z + 8a3z3 - 5a3z5 + a3z7 - a4 + 9a4z2 - 9a4z4 + 2a4z6 - 4a5z + 7a5z3 - 5a5z5 + a5z7 - a6 + 6a6z2 - 5a6z4 + a6z6 |
| V2 and V3, the type 2 and 3 Vassiliev invariants: | {-2, 3} |
|
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 946. 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-18 - q-17 - q-16 + 2q-15 - q-14 - 2q-13 + 2q-12 - q-10 + 2q-9 - 2q-3 + q-1 + q2 |
| 3 | q-36 - q-35 - q-34 + 2q-32 - 2q-30 - q-29 + 2q-28 + q-27 - q-26 + q-24 - q-22 + q-21 + q-20 - 2q-19 - 3q-18 + 3q-17 + 2q-16 - 3q-15 - 4q-14 + 4q-13 + 4q-12 - 3q-11 - 4q-10 + 5q-9 + 5q-8 - 5q-7 - 4q-6 + 4q-5 + 5q-4 - 6q-3 - 4q-2 + 2q-1 + 6 - 3q - 2q2 + 2q4 |
| 4 | q-60 - q-59 - q-58 + 3q-55 - q-54 - q-53 - q-52 - 2q-51 + 4q-50 - 2q-46 + 3q-45 - 2q-44 - q-43 + q-42 + 3q-40 - 2q-39 - 4q-38 - q-37 + q-36 + 6q-35 + q-34 - 6q-33 - 2q-32 - q-31 + 7q-30 + 5q-29 - 5q-28 - 3q-27 - 4q-26 + 5q-25 + 8q-24 - 4q-23 - 3q-22 - 7q-21 + 2q-20 + 10q-19 - 3q-18 - 3q-17 - 8q-16 + q-15 + 12q-14 - q-13 - 3q-12 - 9q-11 + q-10 + 15q-9 - q-8 - 3q-7 - 10q-6 - q-5 + 13q-4 - q-2 - 9q-1 - 2 + 8q + q2 + q3 - 4q4 - 2q5 + 2q6 + q8 |
| 5 | q-90 - q-89 - q-88 + q-85 + 2q-84 - 2q-82 - q-81 - q-80 + 2q-78 + 2q-77 - q-75 - q-74 - q-70 - q-69 + q-68 + 3q-67 + q-66 - q-65 - 3q-64 - 4q-63 - q-62 + 4q-61 + 4q-60 + 4q-59 - q-58 - 6q-57 - 4q-56 + 3q-54 + 6q-53 + 4q-52 - 3q-51 - 4q-50 - 4q-49 - 2q-48 + 2q-47 + 5q-46 + 3q-45 - 4q-43 - 5q-42 - 4q-41 + 2q-40 + 6q-39 + 6q-38 - q-37 - 5q-36 - 8q-35 - 2q-34 + 7q-33 + 10q-32 + 4q-31 - 4q-30 - 12q-29 - 7q-28 + 7q-27 + 13q-26 + 7q-25 - 5q-24 - 15q-23 - 10q-22 + 8q-21 + 14q-20 + 8q-19 - 8q-18 - 16q-17 - 9q-16 + 10q-15 + 16q-14 + 8q-13 - 9q-12 - 17q-11 - 9q-10 + 10q-9 + 18q-8 + 10q-7 - 7q-6 - 17q-5 - 11q-4 + 3q-3 + 16q-2 + 11q-1 - 1 - 11q - 8q2 - 2q3 + 6q4 + 6q5 + 2q6 - 3q7 - 2q8 - 2q9 + 2q12 |
| 6 | q-126 - q-125 - q-124 + q-121 + 3q-119 - q-118 - 2q-117 - q-116 - q-115 - q-113 + 5q-112 - q-108 - q-107 - 3q-106 + 4q-105 - 2q-104 - q-103 + q-101 + 2q-100 + 5q-98 - 3q-97 - 4q-96 - 3q-95 - 3q-94 + q-93 + 3q-92 + 8q-91 + 2q-90 - q-88 - 8q-87 - 4q-86 - 2q-85 + 5q-84 + 2q-83 + 5q-82 + 7q-81 - 5q-80 - 2q-79 - 6q-78 - 2q-77 - 6q-76 + q-75 + 9q-74 + 9q-72 - 2q-70 - 12q-69 - 7q-68 + q-67 - 3q-66 + 16q-65 + 9q-64 + 8q-63 - 10q-62 - 10q-61 - 9q-60 - 13q-59 + 14q-58 + 13q-57 + 17q-56 - q-55 - 6q-54 - 15q-53 - 24q-52 + 5q-51 + 11q-50 + 23q-49 + 8q-48 + 2q-47 - 17q-46 - 32q-45 - 4q-44 + 8q-43 + 27q-42 + 15q-41 + 10q-40 - 18q-39 - 36q-38 - 10q-37 + 5q-36 + 30q-35 + 22q-34 + 15q-33 - 20q-32 - 39q-31 - 13q-30 + 3q-29 + 32q-28 + 25q-27 + 17q-26 - 24q-25 - 41q-24 - 13q-23 + 4q-22 + 33q-21 + 25q-20 + 16q-19 - 27q-18 - 42q-17 - 14q-16 + 6q-15 + 35q-14 + 27q-13 + 18q-12 - 25q-11 - 44q-10 - 19q-9 + 2q-8 + 32q-7 + 30q-6 + 24q-5 - 16q-4 - 37q-3 - 23q-2 - 7q-1 + 19 + 22q + 24q2 - 2q3 - 17q4 - 15q5 - 10q6 + 2q7 + 7q8 + 12q9 + 3q10 - q11 - 4q12 - 4q13 - 2q14 + 2q16 + q18 |
| 7 | q-168 - q-167 - q-166 + q-163 + q-161 + 2q-160 - q-159 - 2q-158 - q-157 - 2q-156 + q-155 + 4q-152 + q-151 - 3q-148 - q-146 - 2q-145 + 2q-144 - q-143 + 2q-141 - 2q-140 + 2q-139 + 3q-138 + q-137 + 2q-136 - 3q-135 - 4q-134 - 2q-133 - 5q-132 - q-131 + 3q-130 + 5q-129 + 8q-128 + 3q-127 - q-126 - 6q-124 - 8q-123 - 4q-122 - 2q-121 + 4q-120 + 4q-119 + 4q-118 + 7q-117 + 4q-116 - 2q-115 - 3q-114 - 7q-113 - 6q-112 - 5q-111 - 5q-110 + 3q-109 + 8q-108 + 7q-107 + 10q-106 + 6q-105 - 2q-104 - 7q-103 - 14q-102 - 12q-101 - 5q-100 + 13q-98 + 18q-97 + 13q-96 + 7q-95 - 5q-94 - 16q-93 - 18q-92 - 17q-91 - 4q-90 + 12q-89 + 17q-88 + 22q-87 + 14q-86 - 2q-85 - 14q-84 - 24q-83 - 22q-82 - 8q-81 + 3q-80 + 22q-79 + 28q-78 + 16q-77 + 5q-76 - 14q-75 - 29q-74 - 26q-73 - 17q-72 + 10q-71 + 28q-70 + 29q-69 + 25q-68 + 4q-67 - 24q-66 - 34q-65 - 35q-64 - 8q-63 + 22q-62 + 32q-61 + 39q-60 + 21q-59 - 13q-58 - 37q-57 - 48q-56 - 25q-55 + 13q-54 + 33q-53 + 49q-52 + 35q-51 - 6q-50 - 39q-49 - 57q-48 - 38q-47 + 7q-46 + 35q-45 + 60q-44 + 44q-43 - 4q-42 - 41q-41 - 63q-40 - 46q-39 + 6q-38 + 40q-37 + 66q-36 + 48q-35 - 6q-34 - 43q-33 - 65q-32 - 48q-31 + 9q-30 + 44q-29 + 67q-28 + 46q-27 - 12q-26 - 44q-25 - 65q-24 - 47q-23 + 10q-22 + 49q-21 + 70q-20 + 46q-19 - 15q-18 - 47q-17 - 69q-16 - 53q-15 + 5q-14 + 52q-13 + 76q-12 + 53q-11 - 4q-10 - 40q-9 - 71q-8 - 63q-7 - 12q-6 + 38q-5 + 69q-4 + 56q-3 + 14q-2 - 17q-1 - 47 - 58q - 27q2 + 11q3 + 40q4 + 37q5 + 19q6 + 6q7 - 15q8 - 29q9 - 20q10 - 4q11 + 8q12 + 12q13 + 8q14 + 6q15 + 2q16 - 5q17 - 4q18 - 4q19 + 2q24 |
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[9, 46]] |
Out[2]= | PD[X[4, 2, 5, 1], X[7, 12, 8, 13], X[10, 3, 11, 4], X[2, 11, 3, 12], > X[5, 14, 6, 15], X[13, 6, 14, 7], X[15, 18, 16, 1], X[9, 17, 10, 16], > X[17, 9, 18, 8]] |
In[3]:= | GaussCode[Knot[9, 46]] |
Out[3]= | GaussCode[1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7] |
In[4]:= | DTCode[Knot[9, 46]] |
Out[4]= | DTCode[4, 10, -14, -12, -16, 2, -6, -18, -8] |
In[5]:= | br = BR[Knot[9, 46]] |
Out[5]= | BR[4, {-1, 2, -1, 2, -3, -2, 1, -2, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 9} |
In[7]:= | BraidIndex[Knot[9, 46]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[9, 46]]] |
| |
Out[8]= | -Graphics- |
In[9]:= | #[Knot[9, 46]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 2, 1, 3, 4, 2} |
In[10]:= | alex = Alexander[Knot[9, 46]][t] |
Out[10]= | 2
5 - - - 2 t
t |
In[11]:= | Conway[Knot[9, 46]][z] |
Out[11]= | 2 1 - 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67],
> Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]} |
In[13]:= | {KnotDet[Knot[9, 46]], KnotSignature[Knot[9, 46]]} |
Out[13]= | {9, 0} |
In[14]:= | Jones[Knot[9, 46]][q] |
Out[14]= | -6 -5 -4 2 -2 1
2 + q - q + q - -- + q - -
3 q
q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[9, 46]} |
In[16]:= | A2Invariant[Knot[9, 46]][q] |
Out[16]= | -20 -18 -12 -10 -8 -6 -2 2 2 + q + q - q - q - q - q + q + 2 q |
In[17]:= | HOMFLYPT[Knot[9, 46]][a, z] |
Out[17]= | 2 4 6 2 2 4 2 2 - a - a + a - a z - a z |
In[18]:= | Kauffman[Knot[9, 46]][a, z] |
Out[18]= | 2 4 6 3 5 2 2 4 2 6 2
2 + a - a - a - 2 a z - 6 a z - 4 a z + 3 a z + 9 a z + 6 a z +
3 3 3 5 3 2 4 4 4 6 4 3 5
> a z + 8 a z + 7 a z - 4 a z - 9 a z - 5 a z - 5 a z -
5 5 2 6 4 6 6 6 3 7 5 7
> 5 a z + a z + 2 a z + a z + a z + a z |
In[19]:= | {Vassiliev[2][Knot[9, 46]], Vassiliev[3][Knot[9, 46]]} |
Out[19]= | {-2, 3} |
In[20]:= | Kh[Knot[9, 46]][q, t] |
Out[20]= | 1 1 1 1 1 1 1 1
- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----
q 13 6 9 5 9 4 7 3 5 3 3 2 3
q t q t q t q t q t q t q t |
In[21]:= | ColouredJones[Knot[9, 46], 2][q] |
Out[21]= | -18 -17 -16 2 -14 2 2 -10 2 2 1 2
q - q - q + --- - q - --- + --- - q + -- - -- + - + q
15 13 12 9 3 q
q q q q q |
| Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 946 |
|
