T(5,4)
From Knot Atlas
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| See other torus knots
Visit T(5,4)'s page at Knotilus! Visit T(5,4)'s page at the original Knot Atlas! |
| Edit T(5,4) Quick Notes
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Edit T(5,4) Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | X17,25,18,24 X10,26,11,25 X3,27,4,26 X11,19,12,18 X4,20,5,19 X27,21,28,20 X5,13,6,12 X28,14,29,13 X21,15,22,14 X29,7,30,6 X22,8,23,7 X15,9,16,8 X23,1,24,30 X16,2,17,1 X9,3,10,2 |
| Gauss code | 14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13 |
| Dowker-Thistlethwaite code | 16 -26 -12 22 -2 -18 28 -8 -24 4 -14 -30 10 -20 -6 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t6−t5 + t2−1 + t−2−t−5 + t−6 |
| Conway polynomial | z12 + 11z10 + 44z8 + 77z6 + 56z4 + 15z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 5, 8 } |
| Jones polynomial | −q13−q11 + q10 + q8 + q6 |
| HOMFLY-PT polynomial (db, data sources) | z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−11z8a−14 + 121z6a−12−45z6a−14 + z6a−16 + 133z4a−12−84z4a−14 + 7z4a−16 + 70z2a−12−70z2a−14 + 15z2a−16 + 14a−12−21a−14 + 9a−16−a−18 |
| Kauffman polynomial (db, data sources) | z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−12z10a−14−11z9a−13−11z9a−15 + 55z8a−12 + 56z8a−14 + z8a−16 + 45z7a−13 + 46z7a−15 + z7a−17−121z6a−12−129z6a−14−8z6a−16−84z5a−13−91z5a−15−7z5a−17 + 133z4a−12 + 154z4a−14 + 21z4a−16 + 70z3a−13 + 84z3a−15 + 14z3a−17−70z2a−12−91z2a−14−22z2a−16−z2a−18−21za−13−28za−15−8za−17−za−19 + 14a−12 + 21a−14 + 9a−16 + a−18 |
| The A2 invariant | Data:T(5,4)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(5,4)/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(5,4)"];
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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]=
| t6−t5 + t2−1 + t−2−t−5 + t−6 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z12 + 11z10 + 44z8 + 77z6 + 56z4 + 15z2 + 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]=
| { 5, 8 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q13−q11 + q10 + q8 + q6 |
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]=
| z12a−12 + 12z10a−12−z10a−14 + 55z8a−12−11z8a−14 + 121z6a−12−45z6a−14 + z6a−16 + 133z4a−12−84z4a−14 + 7z4a−16 + 70z2a−12−70z2a−14 + 15z2a−16 + 14a−12−21a−14 + 9a−16−a−18 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z12a−12 + z12a−14 + z11a−13 + z11a−15−12z10a−12−12z10a−14−11z9a−13−11z9a−15 + 55z8a−12 + 56z8a−14 + z8a−16 + 45z7a−13 + 46z7a−15 + z7a−17−121z6a−12−129z6a−14−8z6a−16−84z5a−13−91z5a−15−7z5a−17 + 133z4a−12 + 154z4a−14 + 21z4a−16 + 70z3a−13 + 84z3a−15 + 14z3a−17−70z2a−12−91z2a−14−22z2a−16−z2a−18−21za−13−28za−15−8za−17−za−19 + 14a−12 + 21a−14 + 9a−16 + a−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(5,4)"];
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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]=
| { t6−t5 + t2−1 + t−2−t−5 + t−6, −q13−q11 + q10 + q8 + q6 } |
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 = 8 is the signature of T(5,4). 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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