T(6,5)
From Knot Atlas
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| See other torus knots
Visit T(6,5)'s page at Knotilus! Visit T(6,5)'s page at the original Knot Atlas! |
| Edit T(6,5) Quick Notes
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Edit T(6,5) Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | X19,29,20,28 X10,30,11,29 X1,31,2,30 X40,32,41,31 X11,21,12,20 X2,22,3,21 X41,23,42,22 X32,24,33,23 X3,13,4,12 X42,14,43,13 X33,15,34,14 X24,16,25,15 X43,5,44,4 X34,6,35,5 X25,7,26,6 X16,8,17,7 X35,45,36,44 X26,46,27,45 X17,47,18,46 X8,48,9,47 X27,37,28,36 X18,38,19,37 X9,39,10,38 X48,40,1,39 |
| Gauss code | -3, -6, -9, 13, 14, 15, 16, -20, -23, -2, -5, 9, 10, 11, 12, -16, -19, -22, -1, 5, 6, 7, 8, -12, -15, -18, -21, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -4, -7, -10, -13, 17, 18, 19, 20, -24 |
| Dowker-Thistlethwaite code | 30 12 -34 -16 38 20 -42 -24 46 28 -2 -32 6 36 -10 -40 14 44 -18 -48 22 4 -26 -8 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t10−t9 + t5−t3 + 1−t−3 + t−5−t−9 + t−10 |
| Conway polynomial | z20 + 19z18 + 152z16 + 665z14 + 1729z12 + 2718z10 + 2518z8 + 1288z6 + 329z4 + 35z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 5, 16 } |
| Jones polynomial | −q19−q17 + q14 + q12 + q10 |
| HOMFLY-PT polynomial (db, data sources) | z20a−20 + 20z18a−20−z18a−22 + 171z16a−20−19z16a−22 + 817z14a−20−153z14a−22 + z14a−24 + 2395z12a−20−681z12a−22 + 15z12a−24 + 4459z10a−20−1833z10a−22 + 92z10a−24 + 5291z8a−20−3069z8a−22 + 297z8a−24−z8a−26 + 3926z6a−20−3169z6a−22 + 540z6a−24−9z6a−26 + 1743z4a−20−1932z4a−22 + 546z4a−24−28z4a−26 + 420z2a−20−630z2a−22 + 280z2a−24−35z2a−26 + 42a−20−84a−22 + 56a−24−14a−26 + a−28 |
| Kauffman polynomial (db, data sources) | Data:T(6,5)/Kauffman Polynomial |
| The A2 invariant | Data:T(6,5)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(6,5)/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(6,5)"];
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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]=
| t10−t9 + t5−t3 + 1−t−3 + t−5−t−9 + t−10 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z20 + 19z18 + 152z16 + 665z14 + 1729z12 + 2718z10 + 2518z8 + 1288z6 + 329z4 + 35z2 + 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, 16 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q19−q17 + q14 + q12 + q10 |
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]=
| z20a−20 + 20z18a−20−z18a−22 + 171z16a−20−19z16a−22 + 817z14a−20−153z14a−22 + z14a−24 + 2395z12a−20−681z12a−22 + 15z12a−24 + 4459z10a−20−1833z10a−22 + 92z10a−24 + 5291z8a−20−3069z8a−22 + 297z8a−24−z8a−26 + 3926z6a−20−3169z6a−22 + 540z6a−24−9z6a−26 + 1743z4a−20−1932z4a−22 + 546z4a−24−28z4a−26 + 420z2a−20−630z2a−22 + 280z2a−24−35z2a−26 + 42a−20−84a−22 + 56a−24−14a−26 + a−28 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| Data:T(6,5)/Kauffman Polynomial |
[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(6,5)"];
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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]=
| { t10−t9 + t5−t3 + 1−t−3 + t−5−t−9 + t−10, −q19−q17 + q14 + q12 + q10 } |
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 = 16 is the signature of T(6,5). 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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