K11a34
From Knot Atlas
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 (Knotscape image) | See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.
Visit K11a34's page at Knotilus! Visit K11a34's page at the original Knot Atlas! |
[edit] Knot presentations
| Planar diagram presentation | X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X10,22,11,21 |
| Gauss code | 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -7, 11, -9 |
| Dowker-Thistlethwaite code | 4 8 14 2 16 18 20 6 22 12 10 |
| 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 + 5t3−14t2 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4 |
| Conway polynomial | −z8−3z6−4z4−2z2 + 1 |
| 2nd Alexander ideal (db, data sources) | {1} |
| Determinant and Signature | { 119, 2 } |
| Jones polynomial | −q8 + 4q7−8q6 + 13q5−17q4 + 19q3−19q2 + 16q−11 + 7q−1−3q−2 + q−3 |
| HOMFLY-PT polynomial (db, data sources) | −z8a−2−6z6a−2 + 2z6a−4 + z6−15z4a−2 + 8z4a−4−z4a−6 + 4z4−17z2a−2 + 11z2a−4−2z2a−6 + 6z2−7a−2 + 5a−4−a−6 + 4 |
| Kauffman polynomial (db, data sources) | z10a−2 + z10a−4 + 4z9a−1 + 8z9a−3 + 4z9a−5 + 12z8a−2 + 14z8a−4 + 7z8a−6 + 5z8 + 3az7−5z7a−1−9z7a−3 + 6z7a−5 + 7z7a−7 + a2z6−41z6a−2−37z6a−4−7z6a−6 + 4z6a−8−14z6−8az5−4z5a−1−9z5a−3−25z5a−5−11z5a−7 + z5a−9−3a2z4 + 53z4a−2 + 39z4a−4−2z4a−6−6z4a−8 + 15z4 + 5az3 + 7z3a−1 + 19z3a−3 + 23z3a−5 + 5z3a−7−z3a−9 + 2a2z2−32z2a−2−20z2a−4 + z2a−6 + 2z2a−8−11z2−az−3za−1−7za−3−7za−5−2za−7 + 7a−2 + 5a−4 + a−6 + 4 |
| The A2 invariant | q8−q6 + 3q4 + 3q−2−5q−4 + 2q−6−3q−8 + 2q−12−2q−14 + 4q−16−q−18 + q−22−q−24 |
| The G2 invariant | q46−2q44 + 5q42−9q40 + 11q38−12q36 + 5q34 + 12q32−34q30 + 62q28−83q26 + 79q24−44q22−30q20 + 134q18−222q16 + 269q14−226q12 + 92q10 + 111q8−317q6 + 451q4−442q2 + 284−18q−2−260q−4 + 446q−6−463q−8 + 316q−10−56q−12−204q−14 + 343q−16−319q−18 + 124q−20 + 141q−22−360q−24 + 430q−26−308q−28 + 18q−30 + 321q−32−592q−34 + 670q−36−522q−38 + 179q−40 + 230q−42−564q−44 + 705q−46−601q−48 + 312q−50 + 52q−52−347q−54 + 467q−56−377q−58 + 147q−60 + 125q−62−297q−64 + 311q−66−157q−68−85q−70 + 313q−72−425q−74 + 382q−76−198q−78−62q−80 + 292q−82−425q−84 + 423q−86−298q−88 + 108q−90 + 81q−92−221q−94 + 271q−96−242q−98 + 159q−100−55q−102−32q−104 + 82q−106−98q−108 + 82q−110−49q−112 + 21q−114 + 3q−116−14q−118 + 15q−120−13q−122 + 7q−124−3q−126 + q−128 |
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["K11a34"];
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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 + 5t3−14t2 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4 |
In[5]:=
| Conway[K][z]
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Out[5]=
| −z8−3z6−4z4−2z2 + 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]=
| −q8 + 4q7−8q6 + 13q5−17q4 + 19q3−19q2 + 16q−11 + 7q−1−3q−2 + q−3 |
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−2−6z6a−2 + 2z6a−4 + z6−15z4a−2 + 8z4a−4−z4a−6 + 4z4−17z2a−2 + 11z2a−4−2z2a−6 + 6z2−7a−2 + 5a−4−a−6 + 4 |
In[10]:=
| Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
| z10a−2 + z10a−4 + 4z9a−1 + 8z9a−3 + 4z9a−5 + 12z8a−2 + 14z8a−4 + 7z8a−6 + 5z8 + 3az7−5z7a−1−9z7a−3 + 6z7a−5 + 7z7a−7 + a2z6−41z6a−2−37z6a−4−7z6a−6 + 4z6a−8−14z6−8az5−4z5a−1−9z5a−3−25z5a−5−11z5a−7 + z5a−9−3a2z4 + 53z4a−2 + 39z4a−4−2z4a−6−6z4a−8 + 15z4 + 5az3 + 7z3a−1 + 19z3a−3 + 23z3a−5 + 5z3a−7−z3a−9 + 2a2z2−32z2a−2−20z2a−4 + z2a−6 + 2z2a−8−11z2−az−3za−1−7za−3−7za−5−2za−7 + 7a−2 + 5a−4 + a−6 + 4 |
[edit] "Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a158,}
Same Jones Polynomial (up to mirroring, 
):
{K11a89,}
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["K11a34"];
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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 + 5t3−14t2 + 25t−29 + 25t−1−14t−2 + 5t−3−t−4, −q8 + 4q7−8q6 + 13q5−17q4 + 19q3−19q2 + 16q−11 + 7q−1−3q−2 + q−3 } |
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]=
| {K11a158,} |
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]=
| {K11a89,} |
[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 K11a34. 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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