K11a163
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a163's page at Knotilus! Visit K11a163's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X10,3,11,4 X18,5,19,6 X14,8,15,7 X16,10,17,9 X2,11,3,12 X20,14,21,13 X8,16,9,15 X22,17,1,18 X12,20,13,19 X6,21,7,22 |
| Gauss code | 1, -6, 2, -1, 3, -11, 4, -8, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 10 18 14 16 2 20 8 22 12 6 |
| A Braid Representative |
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| A Morse Link Presentation | 
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[edit] Three dimensional invariants
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[edit] Four dimensional invariants
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[edit] Polynomial invariants
| Alexander polynomial | −t4 + 6t3−15t2 + 24t−27 + 24t−1−15t−2 + 6t−3−t−4 |
| Conway polynomial | −z8−2z6 + z4 + 2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 119, 2 } |
| Jones polynomial | q7−4q6 + 8q5−13q4 + 17q3−19q2 + 19q−15 + 12q−1−7q−2 + 3q−3−q−4 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−9z4a−2 + 3z4a−4 + 8z4−3a2z2−7z2a−2 + 2z2a−4 + 10z2−2a2−2a−2 + 5 |
| Kauffman polynomial (db, data sources) | 2z10a−2 + 2z10 + 4az9 + 11z9a−1 + 7z9a−3 + 3a2z8 + 12z8a−2 + 11z8a−4 + 4z8 + a3z7−11az7−28z7a−1−5z7a−3 + 11z7a−5−11a2z6−44z6a−2−17z6a−4 + 8z6a−6−30z6−4a3z5 + 4az5 + 13z5a−1−13z5a−3−14z5a−5 + 4z5a−7 + 13a2z4 + 40z4a−2 + 8z4a−4−7z4a−6 + z4a−8 + 37z4 + 5a3z3 + 6az3 + 6z3a−1 + 12z3a−3 + 5z3a−5−2z3a−7−7a2z2−14z2a−2−2z2a−4 + z2a−6−18z2−2a3z−4az−4za−1−2za−3 + 2a2 + 2a−2 + 5 |
| The A2 invariant | −q12−2q6 + 3q4−q2 + 3 + 3q−2−2q−4 + 4q−6−4q−8 + 2q−10−q−12−2q−14 + 2q−16−2q−18 + q−20 |
| The G2 invariant | q60−2q58 + 6q56−11q54 + 15q52−18q50 + 9q48 + 12q46−48q44 + 91q42−123q40 + 113q38−47q36−78q34 + 225q32−330q30 + 336q28−214q26−25q24 + 288q22−482q20 + 515q18−346q16 + 48q14 + 262q12−454q10 + 448q8−255q6−33q4 + 292q2−393 + 310q−2−59q−4−236q−6 + 464q−8−507q−10 + 354q−12−51q−14−304q−16 + 578q−18−666q−20 + 535q−22−213q−24−177q−26 + 493q−28−622q−30 + 507q−32−224q−34−113q−36 + 354q−38−407q−40 + 265q−42−5q−44−237q−46 + 357q−48−301q−50 + 97q−52 + 145q−54−336q−56 + 400q−58−320q−60 + 149q−62 + 55q−64−222q−66 + 302q−68−296q−70 + 215q−72−99q−74−16q−76 + 106q−78−157q−80 + 161q−82−127q−84 + 77q−86−19q−88−26q−90 + 51q−92−61q−94 + 51q−96−32q−98 + 15q−100 + q−102−8q−104 + 10q−106−10q−108 + 6q−110−3q−112 + q−114 |
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["K11a163"];
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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]=
| −t4 + 6t3−15t2 + 24t−27 + 24t−1−15t−2 + 6t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−2z6 + z4 + 2z2 + 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]=
| { 119, 2 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| q7−4q6 + 8q5−13q4 + 17q3−19q2 + 19q−15 + 12q−1−7q−2 + 3q−3−q−4 |
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]=
| −z8a−2−5z6a−2 + z6a−4 + 2z6−a2z4−9z4a−2 + 3z4a−4 + 8z4−3a2z2−7z2a−2 + 2z2a−4 + 10z2−2a2−2a−2 + 5 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| 2z10a−2 + 2z10 + 4az9 + 11z9a−1 + 7z9a−3 + 3a2z8 + 12z8a−2 + 11z8a−4 + 4z8 + a3z7−11az7−28z7a−1−5z7a−3 + 11z7a−5−11a2z6−44z6a−2−17z6a−4 + 8z6a−6−30z6−4a3z5 + 4az5 + 13z5a−1−13z5a−3−14z5a−5 + 4z5a−7 + 13a2z4 + 40z4a−2 + 8z4a−4−7z4a−6 + z4a−8 + 37z4 + 5a3z3 + 6az3 + 6z3a−1 + 12z3a−3 + 5z3a−5−2z3a−7−7a2z2−14z2a−2−2z2a−4 + z2a−6−18z2−2a3z−4az−4za−1−2za−3 + 2a2 + 2a−2 + 5 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a66,}
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["K11a163"];
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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]=
| { −t4 + 6t3−15t2 + 24t−27 + 24t−1−15t−2 + 6t−3−t−4, q7−4q6 + 8q5−13q4 + 17q3−19q2 + 19q−15 + 12q−1−7q−2 + 3q−3−q−4 } |
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]=
| {K11a66,} |
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 = 2 is the signature of K11a163. 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 Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top. |
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