T(10,3)
From Knot Atlas
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| See other torus knots
Visit T(10,3)'s page at Knotilus! Visit T(10,3)'s page at the original Knot Atlas! |
| Edit T(10,3) Quick Notes
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Edit T(10,3) Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | X34,8,35,7 X21,9,22,8 X22,36,23,35 X9,37,10,36 X10,24,11,23 X37,25,38,24 X38,12,39,11 X25,13,26,12 X26,40,27,39 X13,1,14,40 X14,28,15,27 X1,29,2,28 X2,16,3,15 X29,17,30,16 X30,4,31,3 X17,5,18,4 X18,32,19,31 X5,33,6,32 X6,20,7,19 X33,21,34,20 |
| Gauss code | -12, -13, 15, 16, -18, -19, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -1, 3, 4, -6, -7, 9, 10 |
| Dowker-Thistlethwaite code | 28 -30 32 -34 36 -38 40 -2 4 -6 8 -10 12 -14 16 -18 20 -22 24 -26 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9 |
| Conway polynomial | z18 + 17z16 + 119z14 + 443z12 + 946z10 + 1166z8 + 792z6 + 264z4 + 33z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 3, 14 } |
| Jones polynomial | −q20 + q11 + q9 |
| HOMFLY-PT polynomial (db, data sources) | z18a−18 + 18z16a−18−z16a−20 + 136z14a−18−17z14a−20 + 561z12a−18−119z12a−20 + z12a−22 + 1377z10a−18−443z10a−20 + 12z10a−22 + 2057z8a−18−946z8a−20 + 55z8a−22 + 1837z6a−18−1166z6a−20 + 121z6a−22 + 924z4a−18−792z4a−20 + 132z4a−22 + 231z2a−18−264z2a−20 + 66z2a−22 + 22a−18−33a−20 + 12a−22 |
| Kauffman polynomial (db, data sources) | Data:T(10,3)/Kauffman Polynomial |
| The A2 invariant | Data:T(10,3)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(10,3)/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(10,3)"];
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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]=
| t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z18 + 17z16 + 119z14 + 443z12 + 946z10 + 1166z8 + 792z6 + 264z4 + 33z2 + 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, 14 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q20 + q11 + q9 |
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]=
| z18a−18 + 18z16a−18−z16a−20 + 136z14a−18−17z14a−20 + 561z12a−18−119z12a−20 + z12a−22 + 1377z10a−18−443z10a−20 + 12z10a−22 + 2057z8a−18−946z8a−20 + 55z8a−22 + 1837z6a−18−1166z6a−20 + 121z6a−22 + 924z4a−18−792z4a−20 + 132z4a−22 + 231z2a−18−264z2a−20 + 66z2a−22 + 22a−18−33a−20 + 12a−22 |
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(10,3)/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(10,3)"];
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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]=
| { t9−t8 + t6−t5 + t3−t2 + 1−t−2 + t−3−t−5 + t−6−t−8 + t−9, −q20 + q11 + q9 } |
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 = 14 is the signature of T(10,3). 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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