T(16,3)
From Knot Atlas
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| See other torus knots
Visit T(16,3)'s page at Knotilus! Visit T(16,3)'s page at the original Knot Atlas! |
| Edit T(16,3) Quick Notes
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Edit T(16,3) Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | X41,63,42,62 X20,64,21,63 X21,43,22,42 X64,44,1,43 X1,23,2,22 X44,24,45,23 X45,3,46,2 X24,4,25,3 X25,47,26,46 X4,48,5,47 X5,27,6,26 X48,28,49,27 X49,7,50,6 X28,8,29,7 X29,51,30,50 X8,52,9,51 X9,31,10,30 X52,32,53,31 X53,11,54,10 X32,12,33,11 X33,55,34,54 X12,56,13,55 X13,35,14,34 X56,36,57,35 X57,15,58,14 X36,16,37,15 X37,59,38,58 X16,60,17,59 X17,39,18,38 X60,40,61,39 X61,19,62,18 X40,20,41,19 |
| Gauss code | -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, -22, -23, 25, 26, -28, -29, 31, 32, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 23, 24, -26, -27, 29, 30, -32, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -24, -25, 27, 28, -30, -31, 1, 2, -4 |
| Dowker-Thistlethwaite code | 22 -24 26 -28 30 -32 34 -36 38 -40 42 -44 46 -48 50 -52 54 -56 58 -60 62 -64 2 -4 6 -8 10 -12 14 -16 18 -20 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t15−t14 + t12−t11 + t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9−t−11 + t−12−t−14 + t−15 |
| Conway polynomial | z30 + 29z28 + 377z26 + 2901z24 + 14697z22 + 51589z20 + 128593z18 + 229517z16 + 292077z14 + 260729z12 + 158389z10 + 62305z8 + 14637z6 + 1785z4 + 85z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 3, 22 } |
| Jones polynomial | −q32 + q17 + q15 |
| HOMFLY-PT polynomial (db, data sources) | Data:T(16,3)/HOMFLYPT Polynomial |
| Kauffman polynomial (db, data sources) | Data:T(16,3)/Kauffman Polynomial |
| The A2 invariant | Data:T(16,3)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(16,3)/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(16,3)"];
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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]=
| t15−t14 + t12−t11 + t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9−t−11 + t−12−t−14 + t−15 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z30 + 29z28 + 377z26 + 2901z24 + 14697z22 + 51589z20 + 128593z18 + 229517z16 + 292077z14 + 260729z12 + 158389z10 + 62305z8 + 14637z6 + 1785z4 + 85z2 + 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, 22 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q32 + q17 + q15 |
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]=
| Data:T(16,3)/HOMFLYPT Polynomial |
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(16,3)/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(16,3)"];
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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]=
| { t15−t14 + t12−t11 + t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9−t−11 + t−12−t−14 + t−15, −q32 + q17 + q15 } |
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 = 22 is the signature of T(16,3). 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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