T(7,4)
From Knot Atlas
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| See other torus knots
Visit T(7,4)'s page at Knotilus! Visit T(7,4)'s page at the original Knot Atlas! |
| Edit T(7,4) Quick Notes
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Edit T(7,4) Further Notes and Views
[edit] Knot presentations
| Planar diagram presentation | X9,41,10,40 X20,42,21,41 X31,1,32,42 X21,11,22,10 X32,12,33,11 X1,13,2,12 X33,23,34,22 X2,24,3,23 X13,25,14,24 X3,35,4,34 X14,36,15,35 X25,37,26,36 X15,5,16,4 X26,6,27,5 X37,7,38,6 X27,17,28,16 X38,18,39,17 X7,19,8,18 X39,29,40,28 X8,30,9,29 X19,31,20,30 |
| Gauss code | -6, -8, -10, 13, 14, 15, -18, -20, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -2, -4, 7, 8, 9, -12, -14, -16, 19, 20, 21, -3, -5, -7, 10, 11, 12, -15, -17, -19, 1, 2, 3 |
| Dowker-Thistlethwaite code | 12 34 -26 18 40 -32 24 4 -38 30 10 -2 36 16 -8 42 22 -14 6 28 -20 |
| Braid presentation |
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[edit] Polynomial invariants
| Alexander polynomial | t9−t8 + t5−t4 + t2−1 + t−2−t−4 + t−5−t−8 + t−9 |
| Conway polynomial | z18 + 17z16 + 119z14 + 442z12 + 936z10 + 1131z8 + 741z6 + 235z4 + 30z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 7, 14 } |
| Jones polynomial | −q18−q16 + q15−q14 + q13 + q11 + q9 |
| HOMFLY-PT polynomial (db, data sources) | z18a−18 + 18z16a−18−z16a−20 + 136z14a−18−17z14a−20 + 561z12a−18−120z12a−20 + z12a−22 + 1378z10a−18−455z10a−20 + 13z10a−22 + 2067z8a−18−1002z8a−20 + 66z8a−22 + 1873z6a−18−1296z6a−20 + 165z6a−22−z6a−24 + 981z4a−18−951z4a−20 + 211z4a−22−6z4a−24 + 270z2a−18−360z2a−20 + 130z2a−22−10z2a−24 + 30a−18−54a−20 + 30a−22−5a−24 |
| Kauffman polynomial (db, data sources) | Data:T(7,4)/Kauffman Polynomial |
| The A2 invariant | Data:T(7,4)/QuantumInvariant/A2/1,0 |
| The G2 invariant | Data:T(7,4)/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(7,4)"];
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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]=
| t9−t8 + t5−t4 + t2−1 + t−2−t−4 + t−5−t−8 + t−9 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z18 + 17z16 + 119z14 + 442z12 + 936z10 + 1131z8 + 741z6 + 235z4 + 30z2 + 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]=
| { 7, 14 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q18−q16 + q15−q14 + q13 + 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−18−120z12a−20 + z12a−22 + 1378z10a−18−455z10a−20 + 13z10a−22 + 2067z8a−18−1002z8a−20 + 66z8a−22 + 1873z6a−18−1296z6a−20 + 165z6a−22−z6a−24 + 981z4a−18−951z4a−20 + 211z4a−22−6z4a−24 + 270z2a−18−360z2a−20 + 130z2a−22−10z2a−24 + 30a−18−54a−20 + 30a−22−5a−24 |
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(7,4)/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(7,4)"];
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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]=
| { t9−t8 + t5−t4 + t2−1 + t−2−t−4 + t−5−t−8 + t−9, −q18−q16 + q15−q14 + q13 + 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(7,4). 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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