9 16
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Visit 9 16's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 9 16's page at Knotilus! Visit 9 16's page at the original Knot Atlas! |
Knot presentations
| Planar diagram presentation | X4251 X12,4,13,3 X16,6,17,5 X18,8,1,7 X6,18,7,17 X10,16,11,15 X14,10,15,9 X8,14,9,13 X2,12,3,11 |
| Gauss code | 1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4 |
| Dowker-Thistlethwaite code | 4 12 16 18 14 2 8 10 6 |
| Conway Notation | [3,3,2+] |
Length is 10, width is 3. Braid index is 3. |
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Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 9_16.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | |
| 1,1 | |
| 2,0 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | |
| 1,0,0 | |
| 1,0,1 |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | |
| 1,0,0,0 |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | |
| 1,0 |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 |
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Computer Talk
The above data is available with the Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["9 16"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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Out[5]=
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In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 39, 6 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {...}
Same Jones Polynomial (up to mirroring, ): {...}
Vassiliev invariants
| V2 and V3: | (6, 14) |
| V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
| The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 6 is the signature of 9 16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | Not Available |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 29, 2005, 15:27:48)... | |
In[2]:= | PD[Knot[9, 16]] |
Out[2]= | PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[16, 6, 17, 5], X[18, 8, 1, 7],X[6, 18, 7, 17], X[10, 16, 11, 15], X[14, 10, 15, 9],X[8, 14, 9, 13], X[2, 12, 3, 11]] |
In[3]:= | GaussCode[Knot[9, 16]] |
Out[3]= | GaussCode[1, -9, 2, -1, 3, -5, 4, -8, 7, -6, 9, -2, 8, -7, 6, -3, 5, -4] |
In[4]:= | DTCode[Knot[9, 16]] |
Out[4]= | DTCode[4, 12, 16, 18, 14, 2, 8, 10, 6] |
In[5]:= | br = BR[Knot[9, 16]] |
Out[5]= | BR[3, {1, 1, 1, 1, 2, 2, -1, 2, 2, 2}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {3, 10} |
In[7]:= | BraidIndex[Knot[9, 16]] |
Out[7]= | 3 |
In[8]:= | Show[DrawMorseLink[Knot[9, 16]]] |
 | |
| Out[8]= | -Graphics- |
In[9]:= | (#[Knot[9, 16]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 3, 3, 3, {4, 7}, 1} |
In[10]:= | alex = Alexander[Knot[9, 16]][t] |
Out[10]= | 2 5 8 2 3 |
In[11]:= | Conway[Knot[9, 16]][z] |
Out[11]= | 2 4 6 1 + 6 z + 7 z + 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[9, 16]} |
In[13]:= | {KnotDet[Knot[9, 16]], KnotSignature[Knot[9, 16]]} |
Out[13]= | {39, 6} |
In[14]:= | Jones[Knot[9, 16]][q] |
Out[14]= | 3 4 5 6 7 8 9 10 11 12 q - q + 4 q - 5 q + 6 q - 7 q + 6 q - 5 q + 3 q - q |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[9, 16]} |
In[16]:= | A2Invariant[Knot[9, 16]][q] |
Out[16]= | 10 14 16 18 20 22 26 34 36 q + 3 q + q + 2 q + q - 2 q - 3 q + q - q |
In[17]:= | HOMFLYPT[Knot[9, 16]][a, z] |
Out[17]= | 2 2 4 4 4 6 6 |
In[18]:= | Kauffman[Knot[9, 16]][a, z] |
Out[18]= | 2 2 2 2 2 |
In[19]:= | {Vassiliev[2][Knot[9, 16]], Vassiliev[3][Knot[9, 16]]} |
Out[19]= | {6, 14} |
In[20]:= | Kh[Knot[9, 16]][q, t] |
Out[20]= | 5 7 7 9 2 11 2 11 3 13 3 13 4 |
In[21]:= | ColouredJones[Knot[9, 16], 2][q] |
Out[21]= | 6 7 9 10 11 12 13 14 15 |
| See/edit the Rolfsen_Splice_Template.
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