K11a180
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a180's page at Knotilus! Visit K11a180's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X18,9,19,10 X20,11,21,12 X2,13,3,14 X6,16,7,15 X22,18,1,17 X10,19,11,20 X8,21,9,22 |
| Gauss code | 1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -9 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 20 2 6 22 10 8 |
| 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 | t4−5t3 + 11t2−17t + 21−17t−1 + 11t−2−5t−3 + t−4 |
| Conway polynomial | z8 + 3z6 + z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 89, 0 } |
| Jones polynomial | −q5 + 3q4−5q3 + 9q2−12q + 14−14q−1 + 12q−2−9q−3 + 6q−4−3q−5 + q−6 |
| HOMFLY-PT polynomial (db, data sources) | z8−2a2z6−z6a−2 + 6z6 + a4z4−9a2z4−4z4a−2 + 13z4 + 3a4z2−12a2z2−4z2a−2 + 11z2 + 2a4−4a2 + 3 |
| Kauffman polynomial (db, data sources) | a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 4a4z8 + 5a2z8 + 4z8a−2 + 5z8 + 3a5z7−5a3z7−16az7−4z7a−1 + 4z7a−3 + a6z6−11a4z6−22a2z6−7z6a−2 + 3z6a−4−20z6−9a5z5−a3z5 + 18az5 + z5a−1−8z5a−3 + z5a−5−3a6z4 + 8a4z4 + 31a2z4 + 4z4a−2−7z4a−4 + 31z4 + 6a5z3 + a3z3−6az3 + 5z3a−1 + 4z3a−3−2z3a−5 + 2a6z2−5a4z2−20a2z2 + 3z2a−4−16z2−a5z−2za−1−za−3 + 2a4 + 4a2 + 3 |
| The A2 invariant | q18 + q12−2q10 + 2q8−q6−q4 + q2−3 + 3q−2−q−4 + 2q−6 + 2q−8−q−10 + q−12−q−14 |
| The G2 invariant | q94−2q92 + 5q90−9q88 + 10q86−9q84 + 16q80−32q78 + 48q76−53q74 + 38q72−5q70−43q68 + 94q66−122q64 + 121q62−76q60−2q58 + 89q56−155q54 + 177q52−139q50 + 54q48 + 47q46−126q44 + 151q42−112q40 + 33q38 + 56q36−112q34 + 105q32−46q30−52q28 + 139q26−177q24 + 144q22−49q20−78q18 + 186q16−241q14 + 217q12−125q10−9q8 + 131q6−206q4 + 206q2−135 + 32q−2 + 67q−4−120q−6 + 113q−8−52q−10−25q−12 + 91q−14−105q−16 + 68q−18 + 8q−20−85q−22 + 140q−24−141q−26 + 99q−28−27q−30−53q−32 + 108q−34−132q−36 + 119q−38−74q−40 + 21q−42 + 27q−44−62q−46 + 75q−48−70q−50 + 50q−52−23q−54−3q−56 + 22q−58−31q−60 + 30q−62−21q−64 + 13q−66−2q−68−5q−70 + 6q−72−7q−74 + 4q−76−2q−78 + q−80 |
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["K11a180"];
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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]=
| t4−5t3 + 11t2−17t + 21−17t−1 + 11t−2−5t−3 + t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| z8 + 3z6 + 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]=
| { 89, 0 } |
In[8]:=
| Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
| −q5 + 3q4−5q3 + 9q2−12q + 14−14q−1 + 12q−2−9q−3 + 6q−4−3q−5 + q−6 |
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]=
| z8−2a2z6−z6a−2 + 6z6 + a4z4−9a2z4−4z4a−2 + 13z4 + 3a4z2−12a2z2−4z2a−2 + 11z2 + 2a4−4a2 + 3 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 4a4z8 + 5a2z8 + 4z8a−2 + 5z8 + 3a5z7−5a3z7−16az7−4z7a−1 + 4z7a−3 + a6z6−11a4z6−22a2z6−7z6a−2 + 3z6a−4−20z6−9a5z5−a3z5 + 18az5 + z5a−1−8z5a−3 + z5a−5−3a6z4 + 8a4z4 + 31a2z4 + 4z4a−2−7z4a−4 + 31z4 + 6a5z3 + a3z3−6az3 + 5z3a−1 + 4z3a−3−2z3a−5 + 2a6z2−5a4z2−20a2z2 + 3z2a−4−16z2−a5z−2za−1−za−3 + 2a4 + 4a2 + 3 |
[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["K11a180"];
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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]=
| { t4−5t3 + 11t2−17t + 21−17t−1 + 11t−2−5t−3 + t−4, −q5 + 3q4−5q3 + 9q2−12q + 14−14q−1 + 12q−2−9q−3 + 6q−4−3q−5 + q−6 } |
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 = 0 is the signature of K11a180. 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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