T(8,3)
From Knot Atlas
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| See other torus knots
Visit T(8,3)'s page at Knotilus! Visit T(8,3)'s page at the original Knot Atlas! |
| Edit T(8,3) Quick Notes
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Edit T(8,3) Further Notes and Views
![Banco Internacional do Funchal [1]](/wiki/Image:Banif_160.jpg) |
[edit] Knot presentations
| Planar diagram presentation | X11,1,12,32 X22,2,23,1 X23,13,24,12 X2,14,3,13 X3,25,4,24 X14,26,15,25 X15,5,16,4 X26,6,27,5 X27,17,28,16 X6,18,7,17 X7,29,8,28 X18,30,19,29 X19,9,20,8 X30,10,31,9 X31,21,32,20 X10,22,11,21 |
| Gauss code | 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1 |
| Dowker-Thistlethwaite code | 22 -24 26 -28 30 -32 2 -4 6 -8 10 -12 14 -16 18 -20 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t7−t6 + t4−t3 + t−1 + t−1−t−3 + t−4−t−6 + t−7 |
| Conway polynomial | z14 + 13z12 + 65z10 + 157z8 + 189z6 + 105z4 + 21z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 3, 10 } |
| Jones polynomial | −q16 + q9 + q7 |
| HOMFLY-PT polynomial (db, data sources) | z14a−14 + 14z12a−14−z12a−16 + 78z10a−14−13z10a−16 + 221z8a−14−65z8a−16 + z8a−18 + 338z6a−14−157z6a−16 + 8z6a−18 + 273z4a−14−189z4a−16 + 21z4a−18 + 105z2a−14−105z2a−16 + 21z2a−18 + 15a−14−21a−16 + 7a−18 |
| Kauffman polynomial (db, data sources) | z14a−14 + z14a−16 + z13a−15 + z13a−17−14z12a−14−14z12a−16−13z11a−15−13z11a−17 + 78z10a−14 + 78z10a−16 + 65z9a−15 + 65z9a−17−221z8a−14−222z8a−16−z8a−18−157z7a−15−157z7a−17 + 338z6a−14 + 346z6a−16 + 8z6a−18 + 189z5a−15 + 189z5a−17−273z4a−14−294z4a−16−21z4a−18−105z3a−15−105z3a−17 + 105z2a−14 + 126z2a−16 + 21z2a−18 + 21za−15 + 21za−17−15a−14−21a−16−7a−18 |
| The A2 invariant | Data:T(8,3)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(8,3)/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
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]:=
| 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]:=
| K = Knot["T(8,3)"];
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In[4]:=
| Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
| t7−t6 + t4−t3 + t−1 + t−1−t−3 + t−4−t−6 + t−7 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z14 + 13z12 + 65z10 + 157z8 + 189z6 + 105z4 + 21z2 + 1 |
In[6]:=
| 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]=
| {1} |
In[7]:=
| {KnotDet[K], KnotSignature[K]}
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Out[7]=
| { 3, 10 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q16 + q9 + q7 |
In[9]:=
| HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
| z14a−14 + 14z12a−14−z12a−16 + 78z10a−14−13z10a−16 + 221z8a−14−65z8a−16 + z8a−18 + 338z6a−14−157z6a−16 + 8z6a−18 + 273z4a−14−189z4a−16 + 21z4a−18 + 105z2a−14−105z2a−16 + 21z2a−18 + 15a−14−21a−16 + 7a−18 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z14a−14 + z14a−16 + z13a−15 + z13a−17−14z12a−14−14z12a−16−13z11a−15−13z11a−17 + 78z10a−14 + 78z10a−16 + 65z9a−15 + 65z9a−17−221z8a−14−222z8a−16−z8a−18−157z7a−15−157z7a−17 + 338z6a−14 + 346z6a−16 + 8z6a−18 + 189z5a−15 + 189z5a−17−273z4a−14−294z4a−16−21z4a−18−105z3a−15−105z3a−17 + 105z2a−14 + 126z2a−16 + 21z2a−18 + 21za−15 + 21za−17−15a−14−21a−16−7a−18 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, 
):
{}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
| AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
| K = Knot["T(8,3)"];
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In[4]:=
| {A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
| { t7−t6 + t4−t3 + t−1 + t−1−t−3 + t−4−t−6 + t−7, −q16 + q9 + q7 } |
In[5]:=
| DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
| {} |
In[6]:=
| DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
| {} |
[edit] Khovanov Homology
| The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 10 is the signature of T(8,3). 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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[edit] Computer Talk
Much of the above data can be recomputed by Mathematica using the packageKnotTheory`. See A Sample KnotTheory` Session.
[edit] Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top. |
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