10 72
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Visit 10 72's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 72's page at Knotilus! Visit 10 72's page at the original Knot Atlas! |
Knot presentations
| Planar diagram presentation | X4251 X10,6,11,5 X8394 X2,9,3,10 X16,8,17,7 X18,12,19,11 X20,14,1,13 X12,20,13,19 X14,18,15,17 X6,16,7,15 |
| Gauss code | 1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7 |
| Dowker-Thistlethwaite code | 4 8 10 16 2 18 20 6 14 12 |
| Conway Notation | [211,3,2+] |
Length is 11, width is 4. Braid index is 4. |
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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 10_72.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | |
| 1,1 | |
| 2,0 |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | |
| 1,0,0 |
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["10 72"];
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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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{ 73, 4 } |
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: | (2, 4) |
| 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 4 is the signature of 10 72. 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 | Not Available |
| 6 | Not Available |
| 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[10, 72]] |
Out[2]= | PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],X[16, 8, 17, 7], X[18, 12, 19, 11], X[20, 14, 1, 13],X[12, 20, 13, 19], X[14, 18, 15, 17], X[6, 16, 7, 15]] |
In[3]:= | GaussCode[Knot[10, 72]] |
Out[3]= | GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7] |
In[4]:= | DTCode[Knot[10, 72]] |
Out[4]= | DTCode[4, 8, 10, 16, 2, 18, 20, 6, 14, 12] |
In[5]:= | br = BR[Knot[10, 72]] |
Out[5]= | BR[4, {1, 1, 1, 1, 2, 2, -1, 2, -3, 2, -3}] |
In[6]:= | {First[br], Crossings[br]} |
Out[6]= | {4, 11} |
In[7]:= | BraidIndex[Knot[10, 72]] |
Out[7]= | 4 |
In[8]:= | Show[DrawMorseLink[Knot[10, 72]]] |
 | |
| Out[8]= | -Graphics- |
In[9]:= | (#[Knot[10, 72]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex} |
Out[9]= | {Reversible, 2, 3, 3, NotAvailable, 1} |
In[10]:= | alex = Alexander[Knot[10, 72]][t] |
Out[10]= | 2 9 16 2 3 |
In[11]:= | Conway[Knot[10, 72]][z] |
Out[11]= | 2 4 6 1 + 2 z - 3 z - 2 z |
In[12]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[12]= | {Knot[10, 72]} |
In[13]:= | {KnotDet[Knot[10, 72]], KnotSignature[Knot[10, 72]]} |
Out[13]= | {73, 4} |
In[14]:= | Jones[Knot[10, 72]][q] |
Out[14]= | 2 3 4 5 6 7 8 9 |
In[15]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[15]= | {Knot[10, 72]} |
In[16]:= | A2Invariant[Knot[10, 72]][q] |
Out[16]= | 4 6 8 10 12 16 18 20 22 |
In[17]:= | HOMFLYPT[Knot[10, 72]][a, z] |
Out[17]= | 2 2 2 2 4 4 4 4-8 2 2 2 z z 3 z 3 z z 2 z 3 z z |
In[18]:= | Kauffman[Knot[10, 72]][a, z] |
Out[18]= | 2 2 2 2-8 2 2 2 z z 3 z z 2 z 6 z 7 z 8 z |
In[19]:= | {Vassiliev[2][Knot[10, 72]], Vassiliev[3][Knot[10, 72]]} |
Out[19]= | {2, 4} |
In[20]:= | Kh[Knot[10, 72]][q, t] |
Out[20]= | 33 5 1 q q 5 7 7 2 9 2 |
In[21]:= | ColouredJones[Knot[10, 72], 2][q] |
Out[21]= | -2 2 2 3 4 5 6 7 8 |
See/edit the Rolfsen_Splice_Template.
